| L(s) = 1 | − 4·4-s + 10·5-s + 50·9-s − 56·11-s + 16·16-s − 120·19-s − 40·20-s − 25·25-s − 180·29-s + 256·31-s − 200·36-s − 484·41-s + 224·44-s + 500·45-s − 560·55-s − 40·59-s − 1.08e3·61-s − 64·64-s − 2.25e3·71-s + 480·76-s + 1.44e3·79-s + 160·80-s + 1.77e3·81-s − 980·89-s − 1.20e3·95-s − 2.80e3·99-s + 100·100-s + ⋯ |
| L(s) = 1 | − 1/2·4-s + 0.894·5-s + 1.85·9-s − 1.53·11-s + 1/4·16-s − 1.44·19-s − 0.447·20-s − 1/5·25-s − 1.15·29-s + 1.48·31-s − 0.925·36-s − 1.84·41-s + 0.767·44-s + 1.65·45-s − 1.37·55-s − 0.0882·59-s − 2.27·61-s − 1/8·64-s − 3.77·71-s + 0.724·76-s + 2.05·79-s + 0.223·80-s + 2.42·81-s − 1.16·89-s − 1.29·95-s − 2.84·99-s + 1/10·100-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 240100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240100 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.621126635\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.621126635\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2$ | \( 1 + p^{2} T^{2} \) |
| 5 | $C_2$ | \( 1 - 2 p T + p^{3} T^{2} \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 - 50 T^{2} + p^{6} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 28 T + p^{3} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 4250 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 5730 T^{2} + p^{6} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 60 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 20970 T^{2} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 90 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 128 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 45610 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 242 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 27970 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 156570 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 286090 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 20 T + p^{3} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 542 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 413170 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 1128 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 378610 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 720 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 915090 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 490 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 294590 T^{2} + p^{6} T^{4} \) |
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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
−10.59677049081958026467992305107, −10.28310959623063488724665626772, −10.04436436709044549703219310533, −9.336102705172220950322566356252, −9.309622555126909319751524018419, −8.338587278324295044867447015589, −8.250374680932885662051298296313, −7.47186657169440413599139992438, −7.29009788234279204657108503817, −6.51933442468180005342740096683, −6.19042967336081089797820315698, −5.57309162295232177041269287641, −5.04891229892299965853446587551, −4.45588919613803919402442398671, −4.28763849262282265775473743920, −3.36703067994750769363022655717, −2.69679450382151762891425293098, −1.83979462530784620400123479414, −1.59362054481535588627025835858, −0.39010393057023164507448278382,
0.39010393057023164507448278382, 1.59362054481535588627025835858, 1.83979462530784620400123479414, 2.69679450382151762891425293098, 3.36703067994750769363022655717, 4.28763849262282265775473743920, 4.45588919613803919402442398671, 5.04891229892299965853446587551, 5.57309162295232177041269287641, 6.19042967336081089797820315698, 6.51933442468180005342740096683, 7.29009788234279204657108503817, 7.47186657169440413599139992438, 8.250374680932885662051298296313, 8.338587278324295044867447015589, 9.309622555126909319751524018419, 9.336102705172220950322566356252, 10.04436436709044549703219310533, 10.28310959623063488724665626772, 10.59677049081958026467992305107
