| L(s) = 1 | + 0.618·2-s + 0.618·3-s − 1.61·4-s − 1.61·5-s + 0.381·6-s − 4.23·7-s − 2.23·8-s − 2.61·9-s − 1.00·10-s − 0.618·11-s − 1.00·12-s + 13-s − 2.61·14-s − 1.00·15-s + 1.85·16-s + 1.85·17-s − 1.61·18-s + 5.47·19-s + 2.61·20-s − 2.61·21-s − 0.381·22-s + 23-s − 1.38·24-s − 2.38·25-s + 0.618·26-s − 3.47·27-s + 6.85·28-s + ⋯ |
| L(s) = 1 | + 0.437·2-s + 0.356·3-s − 0.809·4-s − 0.723·5-s + 0.155·6-s − 1.60·7-s − 0.790·8-s − 0.872·9-s − 0.316·10-s − 0.186·11-s − 0.288·12-s + 0.277·13-s − 0.699·14-s − 0.258·15-s + 0.463·16-s + 0.449·17-s − 0.381·18-s + 1.25·19-s + 0.585·20-s − 0.571·21-s − 0.0814·22-s + 0.208·23-s − 0.282·24-s − 0.476·25-s + 0.121·26-s − 0.668·27-s + 1.29·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 299 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 299 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 13 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| good | 2 | \( 1 - 0.618T + 2T^{2} \) |
| 3 | \( 1 - 0.618T + 3T^{2} \) |
| 5 | \( 1 + 1.61T + 5T^{2} \) |
| 7 | \( 1 + 4.23T + 7T^{2} \) |
| 11 | \( 1 + 0.618T + 11T^{2} \) |
| 17 | \( 1 - 1.85T + 17T^{2} \) |
| 19 | \( 1 - 5.47T + 19T^{2} \) |
| 29 | \( 1 + 0.145T + 29T^{2} \) |
| 31 | \( 1 + 5.38T + 31T^{2} \) |
| 37 | \( 1 + 4.76T + 37T^{2} \) |
| 41 | \( 1 + 12.7T + 41T^{2} \) |
| 43 | \( 1 - 4.61T + 43T^{2} \) |
| 47 | \( 1 + 12.5T + 47T^{2} \) |
| 53 | \( 1 - 5.76T + 53T^{2} \) |
| 59 | \( 1 + T + 59T^{2} \) |
| 61 | \( 1 + 12.2T + 61T^{2} \) |
| 67 | \( 1 - 4.70T + 67T^{2} \) |
| 71 | \( 1 - 8.70T + 71T^{2} \) |
| 73 | \( 1 + 10.8T + 73T^{2} \) |
| 79 | \( 1 + 13.7T + 79T^{2} \) |
| 83 | \( 1 - 6.85T + 83T^{2} \) |
| 89 | \( 1 - 13.7T + 89T^{2} \) |
| 97 | \( 1 - T + 97T^{2} \) |
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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
−11.57476139287656358269496231833, −10.12879059922706262059545925880, −9.338836623874949156677542962407, −8.544617017536251488680252385056, −7.46182403142790860199329271040, −6.14646729780850484492258147989, −5.18809344061677947364315579497, −3.56694864502391949146914557973, −3.23107760800168814272204396838, 0,
3.23107760800168814272204396838, 3.56694864502391949146914557973, 5.18809344061677947364315579497, 6.14646729780850484492258147989, 7.46182403142790860199329271040, 8.544617017536251488680252385056, 9.338836623874949156677542962407, 10.12879059922706262059545925880, 11.57476139287656358269496231833
