| L(s) = 1 | − 0.618·2-s + 2.61·3-s − 1.61·4-s − 1.61·6-s − 3·7-s + 2.23·8-s + 3.85·9-s + 0.618·11-s − 4.23·12-s − 13-s + 1.85·14-s + 1.85·16-s − 5.23·17-s − 2.38·18-s − 7.85·21-s − 0.381·22-s − 7.61·23-s + 5.85·24-s + 0.618·26-s + 2.23·27-s + 4.85·28-s + 1.38·29-s + 2.14·31-s − 5.61·32-s + 1.61·33-s + 3.23·34-s − 6.23·36-s + ⋯ |
| L(s) = 1 | − 0.437·2-s + 1.51·3-s − 0.809·4-s − 0.660·6-s − 1.13·7-s + 0.790·8-s + 1.28·9-s + 0.186·11-s − 1.22·12-s − 0.277·13-s + 0.495·14-s + 0.463·16-s − 1.26·17-s − 0.561·18-s − 1.71·21-s − 0.0814·22-s − 1.58·23-s + 1.19·24-s + 0.121·26-s + 0.430·27-s + 0.917·28-s + 0.256·29-s + 0.385·31-s − 0.993·32-s + 0.281·33-s + 0.554·34-s − 1.03·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.494563925\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.494563925\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 + 0.618T + 2T^{2} \) |
| 3 | \( 1 - 2.61T + 3T^{2} \) |
| 7 | \( 1 + 3T + 7T^{2} \) |
| 11 | \( 1 - 0.618T + 11T^{2} \) |
| 13 | \( 1 + T + 13T^{2} \) |
| 17 | \( 1 + 5.23T + 17T^{2} \) |
| 23 | \( 1 + 7.61T + 23T^{2} \) |
| 29 | \( 1 - 1.38T + 29T^{2} \) |
| 31 | \( 1 - 2.14T + 31T^{2} \) |
| 37 | \( 1 - 2.14T + 37T^{2} \) |
| 41 | \( 1 - 3T + 41T^{2} \) |
| 43 | \( 1 - 6.85T + 43T^{2} \) |
| 47 | \( 1 + 3T + 47T^{2} \) |
| 53 | \( 1 - 9.32T + 53T^{2} \) |
| 59 | \( 1 - 15.3T + 59T^{2} \) |
| 61 | \( 1 + 5.76T + 61T^{2} \) |
| 67 | \( 1 + 7T + 67T^{2} \) |
| 71 | \( 1 + 1.47T + 71T^{2} \) |
| 73 | \( 1 + 10.7T + 73T^{2} \) |
| 79 | \( 1 - 13.4T + 79T^{2} \) |
| 83 | \( 1 - 0.472T + 83T^{2} \) |
| 89 | \( 1 - 12.2T + 89T^{2} \) |
| 97 | \( 1 - 7.14T + 97T^{2} \) |
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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
−7.85798097625417361571397062123, −7.36280122613455449797895672376, −6.51044743068110102925065968786, −5.80303546096471051127776450172, −4.62016787427499746847381353621, −4.06161454083094428445882318802, −3.48524516652856102137724108632, −2.59139433048905446056033402104, −1.93878368770214206404028467889, −0.56791213225024985668349777347,
0.56791213225024985668349777347, 1.93878368770214206404028467889, 2.59139433048905446056033402104, 3.48524516652856102137724108632, 4.06161454083094428445882318802, 4.62016787427499746847381353621, 5.80303546096471051127776450172, 6.51044743068110102925065968786, 7.36280122613455449797895672376, 7.85798097625417361571397062123
