| L(s) = 1 | − 3.17·2-s − 0.600·3-s + 2.08·4-s + 12.4·5-s + 1.90·6-s + 10.6·7-s + 18.7·8-s − 26.6·9-s − 39.5·10-s + 11·11-s − 1.24·12-s − 33.7·14-s − 7.48·15-s − 76.3·16-s − 62.9·17-s + 84.5·18-s + 140.·19-s + 25.9·20-s − 6.38·21-s − 34.9·22-s + 69.2·23-s − 11.2·24-s + 30.4·25-s + 32.2·27-s + 22.1·28-s − 138.·29-s + 23.7·30-s + ⋯ |
| L(s) = 1 | − 1.12·2-s − 0.115·3-s + 0.260·4-s + 1.11·5-s + 0.129·6-s + 0.573·7-s + 0.830·8-s − 0.986·9-s − 1.25·10-s + 0.301·11-s − 0.0300·12-s − 0.644·14-s − 0.128·15-s − 1.19·16-s − 0.898·17-s + 1.10·18-s + 1.69·19-s + 0.290·20-s − 0.0662·21-s − 0.338·22-s + 0.628·23-s − 0.0959·24-s + 0.243·25-s + 0.229·27-s + 0.149·28-s − 0.885·29-s + 0.144·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 11 | \( 1 - 11T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 3.17T + 8T^{2} \) |
| 3 | \( 1 + 0.600T + 27T^{2} \) |
| 5 | \( 1 - 12.4T + 125T^{2} \) |
| 7 | \( 1 - 10.6T + 343T^{2} \) |
| 17 | \( 1 + 62.9T + 4.91e3T^{2} \) |
| 19 | \( 1 - 140.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 69.2T + 1.21e4T^{2} \) |
| 29 | \( 1 + 138.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 168.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 236.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 207.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 301.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 343.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 306.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 266.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 733.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 180.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 218.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 817.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 289.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 971.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.40e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.72e3T + 9.12e5T^{2} \) |
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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
−8.718023453890415793478516734888, −7.84474205856242746764427945440, −7.14347295766729132249701490153, −6.08571684777395652428379482259, −5.37023996812825594841213939221, −4.56627047624845474629301720699, −3.13609110431426335720096667166, −1.99272234013242907034362134270, −1.23803172475561117724110496300, 0,
1.23803172475561117724110496300, 1.99272234013242907034362134270, 3.13609110431426335720096667166, 4.56627047624845474629301720699, 5.37023996812825594841213939221, 6.08571684777395652428379482259, 7.14347295766729132249701490153, 7.84474205856242746764427945440, 8.718023453890415793478516734888
