| L(s) = 1 | − 2-s + 4-s + 4.11·5-s − 3.78·7-s − 8-s − 4.11·10-s + 2.15·11-s + 5.55·13-s + 3.78·14-s + 16-s − 2.24·17-s + 2.44·19-s + 4.11·20-s − 2.15·22-s − 0.398·23-s + 11.8·25-s − 5.55·26-s − 3.78·28-s + 8.12·29-s + 0.810·31-s − 32-s + 2.24·34-s − 15.5·35-s + 8.17·37-s − 2.44·38-s − 4.11·40-s + 0.726·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s + 1.83·5-s − 1.43·7-s − 0.353·8-s − 1.29·10-s + 0.649·11-s + 1.54·13-s + 1.01·14-s + 0.250·16-s − 0.545·17-s + 0.560·19-s + 0.919·20-s − 0.458·22-s − 0.0831·23-s + 2.37·25-s − 1.08·26-s − 0.715·28-s + 1.50·29-s + 0.145·31-s − 0.176·32-s + 0.385·34-s − 2.62·35-s + 1.34·37-s − 0.396·38-s − 0.649·40-s + 0.113·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8046 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8046 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.215905226\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.215905226\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 149 | \( 1 - T \) |
| good | 5 | \( 1 - 4.11T + 5T^{2} \) |
| 7 | \( 1 + 3.78T + 7T^{2} \) |
| 11 | \( 1 - 2.15T + 11T^{2} \) |
| 13 | \( 1 - 5.55T + 13T^{2} \) |
| 17 | \( 1 + 2.24T + 17T^{2} \) |
| 19 | \( 1 - 2.44T + 19T^{2} \) |
| 23 | \( 1 + 0.398T + 23T^{2} \) |
| 29 | \( 1 - 8.12T + 29T^{2} \) |
| 31 | \( 1 - 0.810T + 31T^{2} \) |
| 37 | \( 1 - 8.17T + 37T^{2} \) |
| 41 | \( 1 - 0.726T + 41T^{2} \) |
| 43 | \( 1 + 3.96T + 43T^{2} \) |
| 47 | \( 1 - 7.08T + 47T^{2} \) |
| 53 | \( 1 + 9.53T + 53T^{2} \) |
| 59 | \( 1 + 11.4T + 59T^{2} \) |
| 61 | \( 1 - 5.54T + 61T^{2} \) |
| 67 | \( 1 - 3.37T + 67T^{2} \) |
| 71 | \( 1 - 8.38T + 71T^{2} \) |
| 73 | \( 1 + 5.85T + 73T^{2} \) |
| 79 | \( 1 + 10.4T + 79T^{2} \) |
| 83 | \( 1 + 1.42T + 83T^{2} \) |
| 89 | \( 1 - 15.6T + 89T^{2} \) |
| 97 | \( 1 + 10.0T + 97T^{2} \) |
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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
−7.961871384434964691945984262384, −6.78575936487506802213817793489, −6.39453672914108574124000846296, −6.15012034604870211688711757881, −5.35550083782714583210874043096, −4.20151038116311911100456919401, −3.17053157999225323626638204549, −2.63539162814471679983821388275, −1.58752221885432939926708736138, −0.879172602861975477086570577663,
0.879172602861975477086570577663, 1.58752221885432939926708736138, 2.63539162814471679983821388275, 3.17053157999225323626638204549, 4.20151038116311911100456919401, 5.35550083782714583210874043096, 6.15012034604870211688711757881, 6.39453672914108574124000846296, 6.78575936487506802213817793489, 7.961871384434964691945984262384
