| L(s) = 1 | − 16.8·2-s + 116.·3-s − 229.·4-s + 1.10e3·5-s − 1.96e3·6-s + 5.16e3·7-s + 1.24e4·8-s − 6.02e3·9-s − 1.85e4·10-s − 4.45e4·11-s − 2.68e4·12-s + 6.96e4·13-s − 8.68e4·14-s + 1.28e5·15-s − 9.20e4·16-s + 1.01e5·18-s − 1.70e5·19-s − 2.53e5·20-s + 6.03e5·21-s + 7.48e5·22-s + 2.00e6·23-s + 1.45e6·24-s − 7.35e5·25-s − 1.17e6·26-s − 3.00e6·27-s − 1.18e6·28-s − 1.55e5·29-s + ⋯ |
| L(s) = 1 | − 0.742·2-s + 0.833·3-s − 0.447·4-s + 0.789·5-s − 0.619·6-s + 0.812·7-s + 1.07·8-s − 0.305·9-s − 0.586·10-s − 0.917·11-s − 0.373·12-s + 0.676·13-s − 0.604·14-s + 0.657·15-s − 0.351·16-s + 0.227·18-s − 0.299·19-s − 0.353·20-s + 0.677·21-s + 0.681·22-s + 1.49·23-s + 0.896·24-s − 0.376·25-s − 0.502·26-s − 1.08·27-s − 0.364·28-s − 0.0408·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 \) |
| good | 2 | \( 1 + 16.8T + 512T^{2} \) |
| 3 | \( 1 - 116.T + 1.96e4T^{2} \) |
| 5 | \( 1 - 1.10e3T + 1.95e6T^{2} \) |
| 7 | \( 1 - 5.16e3T + 4.03e7T^{2} \) |
| 11 | \( 1 + 4.45e4T + 2.35e9T^{2} \) |
| 13 | \( 1 - 6.96e4T + 1.06e10T^{2} \) |
| 19 | \( 1 + 1.70e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 2.00e6T + 1.80e12T^{2} \) |
| 29 | \( 1 + 1.55e5T + 1.45e13T^{2} \) |
| 31 | \( 1 + 4.27e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 1.51e7T + 1.29e14T^{2} \) |
| 41 | \( 1 + 1.59e7T + 3.27e14T^{2} \) |
| 43 | \( 1 - 1.49e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 3.36e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 5.50e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + 7.94e7T + 8.66e15T^{2} \) |
| 61 | \( 1 + 1.27e7T + 1.16e16T^{2} \) |
| 67 | \( 1 - 2.73e8T + 2.72e16T^{2} \) |
| 71 | \( 1 + 3.88e6T + 4.58e16T^{2} \) |
| 73 | \( 1 + 2.32e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + 3.38e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 5.74e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 9.82e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 1.03e9T + 7.60e17T^{2} \) |
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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
−9.492709300393767372760726779174, −8.774474455694329201518181135884, −8.191026205281624651997491165137, −7.27565486784535602799145348811, −5.68122543447105080263082529059, −4.83820848005715224354059452931, −3.45428274531802876411963983976, −2.20229941653023033724735921583, −1.35542484989625708863928133014, 0,
1.35542484989625708863928133014, 2.20229941653023033724735921583, 3.45428274531802876411963983976, 4.83820848005715224354059452931, 5.68122543447105080263082529059, 7.27565486784535602799145348811, 8.191026205281624651997491165137, 8.774474455694329201518181135884, 9.492709300393767372760726779174
