| L(s) = 1 | − 16·2-s − 191.·3-s + 256·4-s − 33.5·5-s + 3.06e3·6-s − 4.09e3·8-s + 1.70e4·9-s + 537.·10-s − 5.15e4·11-s − 4.90e4·12-s − 1.13e5·13-s + 6.44e3·15-s + 6.55e4·16-s + 4.61e5·17-s − 2.73e5·18-s + 8.36e5·19-s − 8.59e3·20-s + 8.24e5·22-s + 2.99e5·23-s + 7.85e5·24-s − 1.95e6·25-s + 1.81e6·26-s + 4.96e5·27-s + 2.58e6·29-s − 1.03e5·30-s + 6.84e6·31-s − 1.04e6·32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.36·3-s + 0.5·4-s − 0.0240·5-s + 0.966·6-s − 0.353·8-s + 0.868·9-s + 0.0169·10-s − 1.06·11-s − 0.683·12-s − 1.10·13-s + 0.0328·15-s + 0.250·16-s + 1.34·17-s − 0.614·18-s + 1.47·19-s − 0.0120·20-s + 0.750·22-s + 0.223·23-s + 0.483·24-s − 0.999·25-s + 0.777·26-s + 0.179·27-s + 0.678·29-s − 0.0232·30-s + 1.33·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + 16T \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + 191.T + 1.96e4T^{2} \) |
| 5 | \( 1 + 33.5T + 1.95e6T^{2} \) |
| 11 | \( 1 + 5.15e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + 1.13e5T + 1.06e10T^{2} \) |
| 17 | \( 1 - 4.61e5T + 1.18e11T^{2} \) |
| 19 | \( 1 - 8.36e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 2.99e5T + 1.80e12T^{2} \) |
| 29 | \( 1 - 2.58e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 6.84e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 1.39e7T + 1.29e14T^{2} \) |
| 41 | \( 1 + 2.22e7T + 3.27e14T^{2} \) |
| 43 | \( 1 + 3.24e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 3.18e6T + 1.11e15T^{2} \) |
| 53 | \( 1 - 8.82e7T + 3.29e15T^{2} \) |
| 59 | \( 1 - 1.39e7T + 8.66e15T^{2} \) |
| 61 | \( 1 - 1.13e8T + 1.16e16T^{2} \) |
| 67 | \( 1 + 3.05e8T + 2.72e16T^{2} \) |
| 71 | \( 1 + 2.86e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + 4.78e7T + 5.88e16T^{2} \) |
| 79 | \( 1 + 1.07e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 5.67e7T + 1.86e17T^{2} \) |
| 89 | \( 1 - 8.48e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 8.99e8T + 7.60e17T^{2} \) |
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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
−11.76074840523807429988582782249, −10.28721007884664262505225904364, −9.881382279972870657892457614052, −8.064486067504777246355922247752, −7.13505186053639836065110722321, −5.77427691864158498442162810874, −4.97156787165134670339516701730, −2.85591614105995915097804527360, −1.06166541950989478253696793333, 0,
1.06166541950989478253696793333, 2.85591614105995915097804527360, 4.97156787165134670339516701730, 5.77427691864158498442162810874, 7.13505186053639836065110722321, 8.064486067504777246355922247752, 9.881382279972870657892457614052, 10.28721007884664262505225904364, 11.76074840523807429988582782249
