| L(s) = 1 | − 2.56·2-s − 1.38·3-s + 4.58·4-s − 5-s + 3.55·6-s − 7-s − 6.64·8-s − 1.08·9-s + 2.56·10-s + 0.594·11-s − 6.35·12-s − 0.500·13-s + 2.56·14-s + 1.38·15-s + 7.87·16-s − 0.443·17-s + 2.78·18-s + 7.44·19-s − 4.58·20-s + 1.38·21-s − 1.52·22-s + 6.10·23-s + 9.19·24-s + 25-s + 1.28·26-s + 5.65·27-s − 4.58·28-s + ⋯ |
| L(s) = 1 | − 1.81·2-s − 0.799·3-s + 2.29·4-s − 0.447·5-s + 1.45·6-s − 0.377·7-s − 2.34·8-s − 0.361·9-s + 0.811·10-s + 0.179·11-s − 1.83·12-s − 0.138·13-s + 0.686·14-s + 0.357·15-s + 1.96·16-s − 0.107·17-s + 0.655·18-s + 1.70·19-s − 1.02·20-s + 0.302·21-s − 0.325·22-s + 1.27·23-s + 1.87·24-s + 0.200·25-s + 0.252·26-s + 1.08·27-s − 0.867·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 229 | \( 1 + T \) |
| good | 2 | \( 1 + 2.56T + 2T^{2} \) |
| 3 | \( 1 + 1.38T + 3T^{2} \) |
| 11 | \( 1 - 0.594T + 11T^{2} \) |
| 13 | \( 1 + 0.500T + 13T^{2} \) |
| 17 | \( 1 + 0.443T + 17T^{2} \) |
| 19 | \( 1 - 7.44T + 19T^{2} \) |
| 23 | \( 1 - 6.10T + 23T^{2} \) |
| 29 | \( 1 - 2.44T + 29T^{2} \) |
| 31 | \( 1 + 10.4T + 31T^{2} \) |
| 37 | \( 1 + 11.1T + 37T^{2} \) |
| 41 | \( 1 + 5.15T + 41T^{2} \) |
| 43 | \( 1 + 2.92T + 43T^{2} \) |
| 47 | \( 1 + 8.93T + 47T^{2} \) |
| 53 | \( 1 - 4.80T + 53T^{2} \) |
| 59 | \( 1 - 2.44T + 59T^{2} \) |
| 61 | \( 1 + 1.25T + 61T^{2} \) |
| 67 | \( 1 - 1.12T + 67T^{2} \) |
| 71 | \( 1 - 11.3T + 71T^{2} \) |
| 73 | \( 1 + 2.53T + 73T^{2} \) |
| 79 | \( 1 - 11.7T + 79T^{2} \) |
| 83 | \( 1 + 15.3T + 83T^{2} \) |
| 89 | \( 1 - 16.7T + 89T^{2} \) |
| 97 | \( 1 - 1.54T + 97T^{2} \) |
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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
−7.48336071474560292338779575762, −6.94160155506109073011122700993, −6.54491521449068185478940020522, −5.46449732824677478857264950431, −5.06726599909046629734931853210, −3.51018559720065605570347193306, −2.98317023106798816546601187647, −1.78209778417067579963929937603, −0.853023548619172176815630530549, 0,
0.853023548619172176815630530549, 1.78209778417067579963929937603, 2.98317023106798816546601187647, 3.51018559720065605570347193306, 5.06726599909046629734931853210, 5.46449732824677478857264950431, 6.54491521449068185478940020522, 6.94160155506109073011122700993, 7.48336071474560292338779575762
