| L(s) = 1 | + 3-s + 4·4-s − 3·5-s + 3·9-s + 4·12-s − 3·15-s + 4·16-s − 12·20-s + 18·23-s + 4·25-s + 8·27-s − 5·31-s + 12·36-s − 14·37-s − 9·45-s − 12·47-s + 4·48-s + 14·49-s − 12·53-s − 15·59-s − 12·60-s − 16·64-s + 13·67-s + 18·69-s + 6·71-s + 4·75-s − 12·80-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 2·4-s − 1.34·5-s + 9-s + 1.15·12-s − 0.774·15-s + 16-s − 2.68·20-s + 3.75·23-s + 4/5·25-s + 1.53·27-s − 0.898·31-s + 2·36-s − 2.30·37-s − 1.34·45-s − 1.75·47-s + 0.577·48-s + 2·49-s − 1.64·53-s − 1.95·59-s − 1.54·60-s − 2·64-s + 1.58·67-s + 2.16·69-s + 0.712·71-s + 0.461·75-s − 1.34·80-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.395979554\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.395979554\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_2^2$ | \( 1 - T - 2 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 11 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 5 | $C_2$$\times$$C_2^2$ | \( ( 1 + 3 T + p T^{2} )^{2}( 1 - 3 T + 4 T^{2} - 3 p T^{3} + p^{2} T^{4} ) \) |
| 7 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 23 | $C_2^2$ | \( ( 1 - 9 T + 58 T^{2} - 9 p T^{3} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2$$\times$$C_2^2$ | \( ( 1 + 5 T + p T^{2} )^{2}( 1 - 5 T - 6 T^{2} - 5 p T^{3} + p^{2} T^{4} ) \) |
| 37 | $C_2^2$ | \( ( 1 + 7 T + 12 T^{2} + 7 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2$$\times$$C_2^2$ | \( ( 1 + 12 T + p T^{2} )^{2}( 1 - 12 T + 97 T^{2} - 12 p T^{3} + p^{2} T^{4} ) \) |
| 53 | $C_2^2$ | \( ( 1 + 6 T - 17 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2$$\times$$C_2^2$ | \( ( 1 + 15 T + p T^{2} )^{2}( 1 - 15 T + 166 T^{2} - 15 p T^{3} + p^{2} T^{4} ) \) |
| 61 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2$$\times$$C_2^2$ | \( ( 1 - 13 T + p T^{2} )^{2}( 1 + 13 T + 102 T^{2} + 13 p T^{3} + p^{2} T^{4} ) \) |
| 71 | $C_2^2$ | \( ( 1 - 3 T - 62 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{4} \) |
| 97 | $C_2$$\times$$C_2^2$ | \( ( 1 - 17 T + p T^{2} )^{2}( 1 + 17 T + 192 T^{2} + 17 p T^{3} + p^{2} T^{4} ) \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.94155766114293925466309290638, −6.88044691679471827077374724403, −6.83264473979086161305483143241, −6.54452840139761737614273450491, −6.54022845390572791204193548925, −6.01177469757137788944813583137, −5.59098006878651892028332741135, −5.57054837717240503792784260084, −5.27040314304921084888147320305, −4.90024826882812434443999556015, −4.70037012907009289712149856883, −4.47292847432876568129754246695, −4.42157411243049579112123416579, −4.00600581249181292045506972857, −3.43189569102355044380200259942, −3.32364863647523442397613885895, −3.32264291956509273171791592512, −3.00929594802718747388762311228, −2.77267209110109941920550044513, −2.26927539528448547007095857670, −2.13357590475621082559304526436, −1.73487086190252880800636592873, −1.26191441579359099278863541245, −1.21564272420255466830522545793, −0.40051643458288703069080122353,
0.40051643458288703069080122353, 1.21564272420255466830522545793, 1.26191441579359099278863541245, 1.73487086190252880800636592873, 2.13357590475621082559304526436, 2.26927539528448547007095857670, 2.77267209110109941920550044513, 3.00929594802718747388762311228, 3.32264291956509273171791592512, 3.32364863647523442397613885895, 3.43189569102355044380200259942, 4.00600581249181292045506972857, 4.42157411243049579112123416579, 4.47292847432876568129754246695, 4.70037012907009289712149856883, 4.90024826882812434443999556015, 5.27040314304921084888147320305, 5.57054837717240503792784260084, 5.59098006878651892028332741135, 6.01177469757137788944813583137, 6.54022845390572791204193548925, 6.54452840139761737614273450491, 6.83264473979086161305483143241, 6.88044691679471827077374724403, 6.94155766114293925466309290638
