| L(s) = 1 | − 1.56·2-s + 8.68·3-s − 5.56·4-s − 3.56·5-s − 13.5·6-s − 27.1·7-s + 21.1·8-s + 48.4·9-s + 5.56·10-s + 15.2·11-s − 48.3·12-s − 13·13-s + 42.4·14-s − 30.9·15-s + 11.4·16-s + 44.5·17-s − 75.6·18-s + 23.9·19-s + 19.8·20-s − 236.·21-s − 23.8·22-s + 122.·23-s + 183.·24-s − 112.·25-s + 20.3·26-s + 186.·27-s + 151.·28-s + ⋯ |
| L(s) = 1 | − 0.552·2-s + 1.67·3-s − 0.695·4-s − 0.318·5-s − 0.922·6-s − 1.46·7-s + 0.935·8-s + 1.79·9-s + 0.175·10-s + 0.418·11-s − 1.16·12-s − 0.277·13-s + 0.810·14-s − 0.532·15-s + 0.178·16-s + 0.635·17-s − 0.990·18-s + 0.289·19-s + 0.221·20-s − 2.45·21-s − 0.230·22-s + 1.11·23-s + 1.56·24-s − 0.898·25-s + 0.153·26-s + 1.32·27-s + 1.02·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.9114306732\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9114306732\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 + 13T \) |
| good | 2 | \( 1 + 1.56T + 8T^{2} \) |
| 3 | \( 1 - 8.68T + 27T^{2} \) |
| 5 | \( 1 + 3.56T + 125T^{2} \) |
| 7 | \( 1 + 27.1T + 343T^{2} \) |
| 11 | \( 1 - 15.2T + 1.33e3T^{2} \) |
| 17 | \( 1 - 44.5T + 4.91e3T^{2} \) |
| 19 | \( 1 - 23.9T + 6.85e3T^{2} \) |
| 23 | \( 1 - 122.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 219.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 27.0T + 2.97e4T^{2} \) |
| 37 | \( 1 - 94.1T + 5.06e4T^{2} \) |
| 41 | \( 1 + 160.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 151.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 466.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 120.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 439.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 137.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 512.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 410.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 308.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 586.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.35e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 439.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.51e3T + 9.12e5T^{2} \) |
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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
−19.30705881950751410189199659154, −18.77683236558196979258764124750, −16.75650718689160499623937765796, −15.24718868092760447281081961144, −13.87746652659443314443937241466, −12.85250954284156864464523605367, −9.827288916794965173808830426038, −9.046158984158336058320394800623, −7.53141004237313528585943938051, −3.56763813396559653157210978973,
3.56763813396559653157210978973, 7.53141004237313528585943938051, 9.046158984158336058320394800623, 9.827288916794965173808830426038, 12.85250954284156864464523605367, 13.87746652659443314443937241466, 15.24718868092760447281081961144, 16.75650718689160499623937765796, 18.77683236558196979258764124750, 19.30705881950751410189199659154
