| L(s) = 1 | − 3.73·2-s − 3·3-s + 5.95·4-s + 3.90·5-s + 11.2·6-s + 7.64·8-s + 9·9-s − 14.5·10-s + 19.1·11-s − 17.8·12-s − 13·13-s − 11.7·15-s − 76.1·16-s + 83.8·17-s − 33.6·18-s − 46.8·19-s + 23.2·20-s − 71.6·22-s + 103.·23-s − 22.9·24-s − 109.·25-s + 48.5·26-s − 27·27-s + 108.·29-s + 43.7·30-s + 147.·31-s + 223.·32-s + ⋯ |
| L(s) = 1 | − 1.32·2-s − 0.577·3-s + 0.744·4-s + 0.349·5-s + 0.762·6-s + 0.337·8-s + 0.333·9-s − 0.461·10-s + 0.526·11-s − 0.429·12-s − 0.277·13-s − 0.201·15-s − 1.19·16-s + 1.19·17-s − 0.440·18-s − 0.565·19-s + 0.260·20-s − 0.694·22-s + 0.941·23-s − 0.195·24-s − 0.877·25-s + 0.366·26-s − 0.192·27-s + 0.693·29-s + 0.266·30-s + 0.854·31-s + 1.23·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 3T \) |
| 7 | \( 1 \) |
| 13 | \( 1 + 13T \) |
| good | 2 | \( 1 + 3.73T + 8T^{2} \) |
| 5 | \( 1 - 3.90T + 125T^{2} \) |
| 11 | \( 1 - 19.1T + 1.33e3T^{2} \) |
| 17 | \( 1 - 83.8T + 4.91e3T^{2} \) |
| 19 | \( 1 + 46.8T + 6.85e3T^{2} \) |
| 23 | \( 1 - 103.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 108.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 147.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 160.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 231.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 340.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 119.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 732.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 229.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 108.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 10.3T + 3.00e5T^{2} \) |
| 71 | \( 1 + 869.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 1.09e3T + 3.89e5T^{2} \) |
| 79 | \( 1 - 140.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 159.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.06e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 858.T + 9.12e5T^{2} \) |
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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
−8.477975783240228225189042310012, −7.87379015322877909871182323826, −6.93557434518788941437290094591, −6.36851349271331545255698142678, −5.28386788146985013319237749891, −4.50232464150569206248988607496, −3.23737000721425224917404462361, −1.86065915165469368728982489823, −1.08363286837777983982960377311, 0,
1.08363286837777983982960377311, 1.86065915165469368728982489823, 3.23737000721425224917404462361, 4.50232464150569206248988607496, 5.28386788146985013319237749891, 6.36851349271331545255698142678, 6.93557434518788941437290094591, 7.87379015322877909871182323826, 8.477975783240228225189042310012
