| L(s) = 1 | − 10·5-s + 50·9-s + 56·11-s − 120·19-s − 25·25-s − 180·29-s + 256·31-s + 484·41-s − 500·45-s + 10·49-s − 560·55-s − 40·59-s + 1.08e3·61-s + 2.25e3·71-s − 1.44e3·79-s + 1.77e3·81-s + 980·89-s + 1.20e3·95-s + 2.80e3·99-s − 1.15e3·101-s − 740·109-s − 310·121-s + 1.50e3·125-s + 127-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 1.85·9-s + 1.53·11-s − 1.44·19-s − 1/5·25-s − 1.15·29-s + 1.48·31-s + 1.84·41-s − 1.65·45-s + 0.0291·49-s − 1.37·55-s − 0.0882·59-s + 2.27·61-s + 3.77·71-s − 2.05·79-s + 2.42·81-s + 1.16·89-s + 1.29·95-s + 2.84·99-s − 1.13·101-s − 0.650·109-s − 0.232·121-s + 1.07·125-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6400 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.930716791\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.930716791\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( 1 + 2 p T + p^{3} T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 50 T^{2} + p^{6} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 10 T^{2} + p^{6} T^{4} \) |
| 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
−14.24871770551616562608790852176, −13.58321738757912432576907864682, −12.78109113831846301638774414288, −12.73159405039226637921851587056, −12.00601385006042567280648410426, −11.48707051583408906738104714388, −10.94615263446431194212458318343, −10.30645565126906576086136323070, −9.581523675084277520018602528974, −9.313257126789916623075637991462, −8.330440342658319081820298780338, −7.920207403941555692963667016788, −7.00360623711879877267743546589, −6.78583866459021974187339627800, −5.97011712467024798075455849164, −4.73074108874878233178853455269, −3.95713950107976570085892972117, −3.91516621417826761121525157483, −2.13212273937316211592754290477, −0.957836559168743372744539639026,
0.957836559168743372744539639026, 2.13212273937316211592754290477, 3.91516621417826761121525157483, 3.95713950107976570085892972117, 4.73074108874878233178853455269, 5.97011712467024798075455849164, 6.78583866459021974187339627800, 7.00360623711879877267743546589, 7.920207403941555692963667016788, 8.330440342658319081820298780338, 9.313257126789916623075637991462, 9.581523675084277520018602528974, 10.30645565126906576086136323070, 10.94615263446431194212458318343, 11.48707051583408906738104714388, 12.00601385006042567280648410426, 12.73159405039226637921851587056, 12.78109113831846301638774414288, 13.58321738757912432576907864682, 14.24871770551616562608790852176
