| L(s) = 1 | + 39.9·2-s − 107.·3-s + 1.08e3·4-s + 181.·5-s − 4.27e3·6-s + 6.42e3·7-s + 2.27e4·8-s − 8.23e3·9-s + 7.26e3·10-s − 1.82e4·11-s − 1.15e5·12-s − 6.43e4·13-s + 2.56e5·14-s − 1.94e4·15-s + 3.55e5·16-s + 3.30e5·17-s − 3.28e5·18-s + 1.96e5·20-s − 6.87e5·21-s − 7.28e5·22-s − 1.73e6·23-s − 2.43e6·24-s − 1.92e6·25-s − 2.57e6·26-s + 2.98e6·27-s + 6.95e6·28-s − 3.22e6·29-s + ⋯ |
| L(s) = 1 | + 1.76·2-s − 0.762·3-s + 2.11·4-s + 0.130·5-s − 1.34·6-s + 1.01·7-s + 1.96·8-s − 0.418·9-s + 0.229·10-s − 0.375·11-s − 1.61·12-s − 0.625·13-s + 1.78·14-s − 0.0992·15-s + 1.35·16-s + 0.960·17-s − 0.737·18-s + 0.275·20-s − 0.771·21-s − 0.662·22-s − 1.29·23-s − 1.49·24-s − 0.983·25-s − 1.10·26-s + 1.08·27-s + 2.13·28-s − 0.846·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 361 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 361 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 19 | \( 1 \) |
| good | 2 | \( 1 - 39.9T + 512T^{2} \) |
| 3 | \( 1 + 107.T + 1.96e4T^{2} \) |
| 5 | \( 1 - 181.T + 1.95e6T^{2} \) |
| 7 | \( 1 - 6.42e3T + 4.03e7T^{2} \) |
| 11 | \( 1 + 1.82e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + 6.43e4T + 1.06e10T^{2} \) |
| 17 | \( 1 - 3.30e5T + 1.18e11T^{2} \) |
| 23 | \( 1 + 1.73e6T + 1.80e12T^{2} \) |
| 29 | \( 1 + 3.22e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 6.34e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 1.71e7T + 1.29e14T^{2} \) |
| 41 | \( 1 - 8.37e6T + 3.27e14T^{2} \) |
| 43 | \( 1 + 3.17e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 3.65e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 5.01e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + 2.30e7T + 8.66e15T^{2} \) |
| 61 | \( 1 - 1.33e8T + 1.16e16T^{2} \) |
| 67 | \( 1 + 3.98e7T + 2.72e16T^{2} \) |
| 71 | \( 1 + 4.02e8T + 4.58e16T^{2} \) |
| 73 | \( 1 - 3.15e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 2.13e7T + 1.19e17T^{2} \) |
| 83 | \( 1 + 4.17e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 8.58e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 8.94e8T + 7.60e17T^{2} \) |
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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
−9.792047246755828256810388142167, −8.096942425991584820253039144188, −7.28980164660984968664804795344, −5.95659568301585226785968411981, −5.56663275225006119391733790970, −4.75780358116027377455875322177, −3.79902619549945844475983202253, −2.57844358202122734357350936039, −1.62963633813865536994040070939, 0,
1.62963633813865536994040070939, 2.57844358202122734357350936039, 3.79902619549945844475983202253, 4.75780358116027377455875322177, 5.56663275225006119391733790970, 5.95659568301585226785968411981, 7.28980164660984968664804795344, 8.096942425991584820253039144188, 9.792047246755828256810388142167
