| L(s) = 1 | + 1.00·2-s − 3.02·3-s − 6.99·4-s − 20.0·5-s − 3.03·6-s + 1.63·7-s − 15.0·8-s − 17.8·9-s − 20.0·10-s + 11·11-s + 21.1·12-s + 1.63·14-s + 60.7·15-s + 40.9·16-s − 84.4·17-s − 17.8·18-s − 92.6·19-s + 140.·20-s − 4.95·21-s + 11.0·22-s + 18.2·23-s + 45.5·24-s + 276.·25-s + 135.·27-s − 11.4·28-s + 81.2·29-s + 60.8·30-s + ⋯ |
| L(s) = 1 | + 0.354·2-s − 0.583·3-s − 0.874·4-s − 1.79·5-s − 0.206·6-s + 0.0883·7-s − 0.663·8-s − 0.660·9-s − 0.634·10-s + 0.301·11-s + 0.509·12-s + 0.0312·14-s + 1.04·15-s + 0.639·16-s − 1.20·17-s − 0.233·18-s − 1.11·19-s + 1.56·20-s − 0.0514·21-s + 0.106·22-s + 0.165·23-s + 0.387·24-s + 2.21·25-s + 0.967·27-s − 0.0772·28-s + 0.520·29-s + 0.370·30-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 | \( 1 - 1.00T + 8T^{2} \) |
| 3 | \( 1 + 3.02T + 27T^{2} \) |
| 5 | \( 1 + 20.0T + 125T^{2} \) |
| 7 | \( 1 - 1.63T + 343T^{2} \) |
| 17 | \( 1 + 84.4T + 4.91e3T^{2} \) |
| 19 | \( 1 + 92.6T + 6.85e3T^{2} \) |
| 23 | \( 1 - 18.2T + 1.21e4T^{2} \) |
| 29 | \( 1 - 81.2T + 2.43e4T^{2} \) |
| 31 | \( 1 - 129.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 305.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 271.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 7.78T + 7.95e4T^{2} \) |
| 47 | \( 1 - 249.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 284.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 571.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 435.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 465.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 667.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 661.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 857.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 97.4T + 5.71e5T^{2} \) |
| 89 | \( 1 - 563.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.76e3T + 9.12e5T^{2} \) |
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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
−8.583155657309691030872007940136, −7.85525385651859778413396323747, −6.77737043397550598448833180892, −6.08933182451565391138267179625, −4.89243950336458295589964344857, −4.44031428280410560553553521965, −3.72614228809736880840007463627, −2.72009917523882280410580388979, −0.73429454753725118047108998826, 0,
0.73429454753725118047108998826, 2.72009917523882280410580388979, 3.72614228809736880840007463627, 4.44031428280410560553553521965, 4.89243950336458295589964344857, 6.08933182451565391138267179625, 6.77737043397550598448833180892, 7.85525385651859778413396323747, 8.583155657309691030872007940136
