| L(s) = 1 | + 2-s + 4-s + 0.215·5-s + 2.46·7-s + 8-s + 0.215·10-s + 4.52·11-s − 0.00998·13-s + 2.46·14-s + 16-s − 5.54·17-s − 2.56·19-s + 0.215·20-s + 4.52·22-s + 2.00·23-s − 4.95·25-s − 0.00998·26-s + 2.46·28-s + 7.30·29-s + 8.98·31-s + 32-s − 5.54·34-s + 0.531·35-s − 3.67·37-s − 2.56·38-s + 0.215·40-s + 2.53·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s + 0.0963·5-s + 0.932·7-s + 0.353·8-s + 0.0680·10-s + 1.36·11-s − 0.00276·13-s + 0.659·14-s + 0.250·16-s − 1.34·17-s − 0.588·19-s + 0.0481·20-s + 0.965·22-s + 0.417·23-s − 0.990·25-s − 0.00195·26-s + 0.466·28-s + 1.35·29-s + 1.61·31-s + 0.176·32-s − 0.950·34-s + 0.0898·35-s − 0.604·37-s − 0.415·38-s + 0.0340·40-s + 0.396·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6354 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.121829558\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.121829558\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 353 | \( 1 - T \) |
| good | 5 | \( 1 - 0.215T + 5T^{2} \) |
| 7 | \( 1 - 2.46T + 7T^{2} \) |
| 11 | \( 1 - 4.52T + 11T^{2} \) |
| 13 | \( 1 + 0.00998T + 13T^{2} \) |
| 17 | \( 1 + 5.54T + 17T^{2} \) |
| 19 | \( 1 + 2.56T + 19T^{2} \) |
| 23 | \( 1 - 2.00T + 23T^{2} \) |
| 29 | \( 1 - 7.30T + 29T^{2} \) |
| 31 | \( 1 - 8.98T + 31T^{2} \) |
| 37 | \( 1 + 3.67T + 37T^{2} \) |
| 41 | \( 1 - 2.53T + 41T^{2} \) |
| 43 | \( 1 - 3.41T + 43T^{2} \) |
| 47 | \( 1 - 4.67T + 47T^{2} \) |
| 53 | \( 1 - 10.1T + 53T^{2} \) |
| 59 | \( 1 + 8.13T + 59T^{2} \) |
| 61 | \( 1 - 0.423T + 61T^{2} \) |
| 67 | \( 1 - 15.2T + 67T^{2} \) |
| 71 | \( 1 - 12.8T + 71T^{2} \) |
| 73 | \( 1 - 6.76T + 73T^{2} \) |
| 79 | \( 1 + 9.88T + 79T^{2} \) |
| 83 | \( 1 + 6.14T + 83T^{2} \) |
| 89 | \( 1 + 5.17T + 89T^{2} \) |
| 97 | \( 1 + 16.8T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.161720876969061516862426627443, −7.02559551742861939826608597758, −6.61886413690134898102401444024, −5.94320297029924196314422521365, −5.01489276115725058365513112457, −4.33720834968260950790971619663, −3.94545210042605508971567385265, −2.70856897997046540788729344708, −1.96830479203507207108153054716, −1.00384024247237646914939839451,
1.00384024247237646914939839451, 1.96830479203507207108153054716, 2.70856897997046540788729344708, 3.94545210042605508971567385265, 4.33720834968260950790971619663, 5.01489276115725058365513112457, 5.94320297029924196314422521365, 6.61886413690134898102401444024, 7.02559551742861939826608597758, 8.161720876969061516862426627443
