| L(s) = 1 | − 14·5-s − 17·9-s − 32·11-s − 28·13-s − 154·17-s + 224·19-s − 68·23-s + 45·25-s − 236·29-s + 196·31-s + 346·37-s − 420·41-s + 344·43-s + 238·45-s − 84·47-s − 438·53-s + 448·55-s + 56·59-s + 98·61-s + 392·65-s − 336·67-s − 896·71-s + 966·73-s + 52·79-s − 440·81-s − 392·83-s + 2.15e3·85-s + ⋯ |
| L(s) = 1 | − 1.25·5-s − 0.629·9-s − 0.877·11-s − 0.597·13-s − 2.19·17-s + 2.70·19-s − 0.616·23-s + 9/25·25-s − 1.51·29-s + 1.13·31-s + 1.53·37-s − 1.59·41-s + 1.21·43-s + 0.788·45-s − 0.260·47-s − 1.13·53-s + 1.09·55-s + 0.123·59-s + 0.205·61-s + 0.748·65-s − 0.612·67-s − 1.49·71-s + 1.54·73-s + 0.0740·79-s − 0.603·81-s − 0.518·83-s + 2.75·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 614656 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 614656 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 + 17 T^{2} + p^{6} T^{4} \) |
| 5 | $D_{4}$ | \( 1 + 14 T + 151 T^{2} + 14 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 32 T + 1105 T^{2} + 32 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 28 T + 3998 T^{2} + 28 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 154 T + 15163 T^{2} + 154 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 224 T + 25929 T^{2} - 224 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 68 T + 23677 T^{2} + 68 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 236 T + 33694 T^{2} + 236 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 196 T + 67373 T^{2} - 196 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 346 T + 65967 T^{2} - 346 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 420 T + 176614 T^{2} + 420 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 4 p T + p^{3} T^{2} )^{2} \) |
| 47 | $D_{4}$ | \( 1 + 84 T + 201085 T^{2} + 84 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 438 T + 280447 T^{2} + 438 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 56 T + 360889 T^{2} - 56 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 98 T + 253751 T^{2} - 98 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 336 T + 627937 T^{2} + 336 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 896 T + 800494 T^{2} + 896 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 966 T + 187 p^{2} T^{2} - 966 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 52 T + 332261 T^{2} - 52 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 392 T + 895462 T^{2} + 392 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 294 T + 1298347 T^{2} - 294 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 420 T + 1655734 T^{2} + 420 p^{3} T^{3} + p^{6} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.645261181769491518885162707720, −9.450424875295049802739073473235, −8.563654133035624505600337907337, −8.539407360589365635377317724986, −7.80449002203200132724034997362, −7.65064660121045645297549649191, −7.22820377949187918277292386494, −6.80992660526513325483475109735, −6.04096884627097917129064406904, −5.76264508631395110327849998206, −5.04304511368037987862711014957, −4.76825383697007177605592223196, −4.20416483457064614251192137781, −3.67341404786471203918467037914, −3.03359448191211918568964883698, −2.66634509477545604428909428369, −1.98379850610639635279080954549, −1.00517661319944208336295671969, 0, 0,
1.00517661319944208336295671969, 1.98379850610639635279080954549, 2.66634509477545604428909428369, 3.03359448191211918568964883698, 3.67341404786471203918467037914, 4.20416483457064614251192137781, 4.76825383697007177605592223196, 5.04304511368037987862711014957, 5.76264508631395110327849998206, 6.04096884627097917129064406904, 6.80992660526513325483475109735, 7.22820377949187918277292386494, 7.65064660121045645297549649191, 7.80449002203200132724034997362, 8.539407360589365635377317724986, 8.563654133035624505600337907337, 9.450424875295049802739073473235, 9.645261181769491518885162707720
