| L(s) = 1 | − 0.278·3-s − 0.533·5-s + 1.93·7-s − 2.92·9-s − 3.65·11-s + 5.69·13-s + 0.148·15-s + 6.46·17-s − 4.37·19-s − 0.540·21-s − 8.89·23-s − 4.71·25-s + 1.65·27-s − 0.939·29-s + 6.67·31-s + 1.01·33-s − 1.03·35-s + 0.254·37-s − 1.58·39-s + 1.94·41-s + 1.83·43-s + 1.55·45-s − 5.30·47-s − 3.24·49-s − 1.80·51-s + 1.23·53-s + 1.94·55-s + ⋯ |
| L(s) = 1 | − 0.160·3-s − 0.238·5-s + 0.732·7-s − 0.974·9-s − 1.10·11-s + 1.58·13-s + 0.0383·15-s + 1.56·17-s − 1.00·19-s − 0.117·21-s − 1.85·23-s − 0.943·25-s + 0.317·27-s − 0.174·29-s + 1.19·31-s + 0.177·33-s − 0.174·35-s + 0.0419·37-s − 0.254·39-s + 0.304·41-s + 0.279·43-s + 0.232·45-s − 0.774·47-s − 0.463·49-s − 0.252·51-s + 0.170·53-s + 0.262·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 503 | \( 1 + T \) |
| good | 3 | \( 1 + 0.278T + 3T^{2} \) |
| 5 | \( 1 + 0.533T + 5T^{2} \) |
| 7 | \( 1 - 1.93T + 7T^{2} \) |
| 11 | \( 1 + 3.65T + 11T^{2} \) |
| 13 | \( 1 - 5.69T + 13T^{2} \) |
| 17 | \( 1 - 6.46T + 17T^{2} \) |
| 19 | \( 1 + 4.37T + 19T^{2} \) |
| 23 | \( 1 + 8.89T + 23T^{2} \) |
| 29 | \( 1 + 0.939T + 29T^{2} \) |
| 31 | \( 1 - 6.67T + 31T^{2} \) |
| 37 | \( 1 - 0.254T + 37T^{2} \) |
| 41 | \( 1 - 1.94T + 41T^{2} \) |
| 43 | \( 1 - 1.83T + 43T^{2} \) |
| 47 | \( 1 + 5.30T + 47T^{2} \) |
| 53 | \( 1 - 1.23T + 53T^{2} \) |
| 59 | \( 1 + 1.09T + 59T^{2} \) |
| 61 | \( 1 + 12.6T + 61T^{2} \) |
| 67 | \( 1 - 11.3T + 67T^{2} \) |
| 71 | \( 1 + 9.49T + 71T^{2} \) |
| 73 | \( 1 + 0.549T + 73T^{2} \) |
| 79 | \( 1 + 1.46T + 79T^{2} \) |
| 83 | \( 1 - 2.74T + 83T^{2} \) |
| 89 | \( 1 - 8.78T + 89T^{2} \) |
| 97 | \( 1 + 8.40T + 97T^{2} \) |
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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
−8.103019089042287399858425668628, −7.74450660712994339893665469946, −6.30395818341599697255012024215, −5.91011100446951310688855451268, −5.21094894878429743914758099153, −4.21963931410487702706484971520, −3.45760690358558678214078626548, −2.47781659610633706560851268263, −1.40375186327561212637203466500, 0,
1.40375186327561212637203466500, 2.47781659610633706560851268263, 3.45760690358558678214078626548, 4.21963931410487702706484971520, 5.21094894878429743914758099153, 5.91011100446951310688855451268, 6.30395818341599697255012024215, 7.74450660712994339893665469946, 8.103019089042287399858425668628
