| L(s) = 1 | − 9·3-s − 4·5-s − 6·7-s + 54·9-s − 32·11-s + 39·13-s + 36·15-s + 158·17-s − 70·19-s + 54·21-s − 176·23-s − 75·25-s − 270·27-s + 222·29-s − 54·31-s + 288·33-s + 24·35-s − 90·37-s − 351·39-s + 104·41-s − 140·43-s − 216·45-s − 328·47-s − 529·49-s − 1.42e3·51-s − 358·53-s + 128·55-s + ⋯ |
| L(s) = 1 | − 1.73·3-s − 0.357·5-s − 0.323·7-s + 2·9-s − 0.877·11-s + 0.832·13-s + 0.619·15-s + 2.25·17-s − 0.845·19-s + 0.561·21-s − 1.59·23-s − 3/5·25-s − 1.92·27-s + 1.42·29-s − 0.312·31-s + 1.51·33-s + 0.115·35-s − 0.399·37-s − 1.44·39-s + 0.396·41-s − 0.496·43-s − 0.715·45-s − 1.01·47-s − 1.54·49-s − 3.90·51-s − 0.927·53-s + 0.313·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{3} \cdot 13^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{3} \cdot 13^{3}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 + p T )^{3} \) |
| 13 | $C_1$ | \( ( 1 - p T )^{3} \) |
| good | 5 | $S_4\times C_2$ | \( 1 + 4 T + 91 T^{2} + 1832 T^{3} + 91 p^{3} T^{4} + 4 p^{6} T^{5} + p^{9} T^{6} \) |
| 7 | $S_4\times C_2$ | \( 1 + 6 T + 565 T^{2} + 3284 T^{3} + 565 p^{3} T^{4} + 6 p^{6} T^{5} + p^{9} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 + 32 T + 4073 T^{2} + 82880 T^{3} + 4073 p^{3} T^{4} + 32 p^{6} T^{5} + p^{9} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 - 158 T + 13055 T^{2} - 792804 T^{3} + 13055 p^{3} T^{4} - 158 p^{6} T^{5} + p^{9} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 + 70 T + 75 p T^{2} - 353052 T^{3} + 75 p^{4} T^{4} + 70 p^{6} T^{5} + p^{9} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 + 176 T + 36821 T^{2} + 3518880 T^{3} + 36821 p^{3} T^{4} + 176 p^{6} T^{5} + p^{9} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 - 222 T + 72459 T^{2} - 9941684 T^{3} + 72459 p^{3} T^{4} - 222 p^{6} T^{5} + p^{9} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 + 54 T + 18189 T^{2} + 7655380 T^{3} + 18189 p^{3} T^{4} + 54 p^{6} T^{5} + p^{9} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 + 90 T + 144243 T^{2} + 9094588 T^{3} + 144243 p^{3} T^{4} + 90 p^{6} T^{5} + p^{9} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 - 104 T + 145119 T^{2} - 16575520 T^{3} + 145119 p^{3} T^{4} - 104 p^{6} T^{5} + p^{9} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 + 140 T + 164713 T^{2} + 21143304 T^{3} + 164713 p^{3} T^{4} + 140 p^{6} T^{5} + p^{9} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 + 328 T + 99661 T^{2} + 12914800 T^{3} + 99661 p^{3} T^{4} + 328 p^{6} T^{5} + p^{9} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 + 358 T + 240563 T^{2} + 84086500 T^{3} + 240563 p^{3} T^{4} + 358 p^{6} T^{5} + p^{9} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 + 1060 T + 964313 T^{2} + 468555544 T^{3} + 964313 p^{3} T^{4} + 1060 p^{6} T^{5} + p^{9} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 + 186 T + 627403 T^{2} + 75772828 T^{3} + 627403 p^{3} T^{4} + 186 p^{6} T^{5} + p^{9} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 + 354 T + 291073 T^{2} + 106732108 T^{3} + 291073 p^{3} T^{4} + 354 p^{6} T^{5} + p^{9} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 + 1692 T + 1393989 T^{2} + 857010312 T^{3} + 1393989 p^{3} T^{4} + 1692 p^{6} T^{5} + p^{9} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 + 974 T + 1376391 T^{2} + 760749700 T^{3} + 1376391 p^{3} T^{4} + 974 p^{6} T^{5} + p^{9} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 + 776 T + 951245 T^{2} + 363935984 T^{3} + 951245 p^{3} T^{4} + 776 p^{6} T^{5} + p^{9} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 + 2340 T + 3482545 T^{2} + 3107827032 T^{3} + 3482545 p^{3} T^{4} + 2340 p^{6} T^{5} + p^{9} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 + 684 T + 2141247 T^{2} + 940696216 T^{3} + 2141247 p^{3} T^{4} + 684 p^{6} T^{5} + p^{9} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 + 742 T - 166337 T^{2} - 354304524 T^{3} - 166337 p^{3} T^{4} + 742 p^{6} T^{5} + p^{9} T^{6} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.726269190880028284519500884620, −9.293218749784898745965609821990, −8.991295766749677425617197667542, −8.421177813158901596650689205879, −8.103430200097005327058018624256, −8.079800066106881357114366719677, −7.79027363598270318609980440252, −7.32444335473090235270531615879, −7.07639955972183165464598907718, −6.71681765919140209181995452298, −6.22855775692313797677855757660, −6.13633467934958653813781163805, −5.84049468002622530892331759958, −5.61335225456098238290590488733, −5.24335654653058596631574989829, −4.89882587421777491316477678880, −4.33266071353213722958461530170, −4.28071552216738719142413551418, −3.94550600104159854951898584534, −3.15931154897823617709911814459, −3.10057993768717089241298325233, −2.70202224438397554846499407021, −1.64850874496815260019301476871, −1.42187229870014565175242963864, −1.28545451547438801563788370595, 0, 0, 0,
1.28545451547438801563788370595, 1.42187229870014565175242963864, 1.64850874496815260019301476871, 2.70202224438397554846499407021, 3.10057993768717089241298325233, 3.15931154897823617709911814459, 3.94550600104159854951898584534, 4.28071552216738719142413551418, 4.33266071353213722958461530170, 4.89882587421777491316477678880, 5.24335654653058596631574989829, 5.61335225456098238290590488733, 5.84049468002622530892331759958, 6.13633467934958653813781163805, 6.22855775692313797677855757660, 6.71681765919140209181995452298, 7.07639955972183165464598907718, 7.32444335473090235270531615879, 7.79027363598270318609980440252, 8.079800066106881357114366719677, 8.103430200097005327058018624256, 8.421177813158901596650689205879, 8.991295766749677425617197667542, 9.293218749784898745965609821990, 9.726269190880028284519500884620
