| L(s) = 1 | − 2·2-s − 3·3-s + 4·4-s + 13.8·5-s + 6·6-s − 21.4·7-s − 8·8-s + 9·9-s − 27.7·10-s + 30.6·11-s − 12·12-s − 50.3·13-s + 42.8·14-s − 41.6·15-s + 16·16-s + 112.·17-s − 18·18-s − 81.3·19-s + 55.5·20-s + 64.2·21-s − 61.3·22-s + 72.0·23-s + 24·24-s + 67.5·25-s + 100.·26-s − 27·27-s − 85.7·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s + 1.24·5-s + 0.408·6-s − 1.15·7-s − 0.353·8-s + 0.333·9-s − 0.877·10-s + 0.840·11-s − 0.288·12-s − 1.07·13-s + 0.818·14-s − 0.716·15-s + 0.250·16-s + 1.60·17-s − 0.235·18-s − 0.982·19-s + 0.620·20-s + 0.668·21-s − 0.594·22-s + 0.653·23-s + 0.204·24-s + 0.540·25-s + 0.758·26-s − 0.192·27-s − 0.578·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 786 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 786 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + 2T \) |
| 3 | \( 1 + 3T \) |
| 131 | \( 1 - 131T \) |
| good | 5 | \( 1 - 13.8T + 125T^{2} \) |
| 7 | \( 1 + 21.4T + 343T^{2} \) |
| 11 | \( 1 - 30.6T + 1.33e3T^{2} \) |
| 13 | \( 1 + 50.3T + 2.19e3T^{2} \) |
| 17 | \( 1 - 112.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 81.3T + 6.85e3T^{2} \) |
| 23 | \( 1 - 72.0T + 1.21e4T^{2} \) |
| 29 | \( 1 + 239.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 266.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 133.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 269.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 21.7T + 7.95e4T^{2} \) |
| 47 | \( 1 + 11.2T + 1.03e5T^{2} \) |
| 53 | \( 1 - 232.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 38.4T + 2.05e5T^{2} \) |
| 61 | \( 1 - 368.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 319.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 722.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 660.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 151.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 750.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 924.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 767.T + 9.12e5T^{2} \) |
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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
−9.541213078910765803227280733028, −9.086107904753971079322501010284, −7.56732007923316637212302928197, −6.83289301392679131193539115649, −5.97736110289531596964592855397, −5.41830921521966014793595626320, −3.79773994986688104433043088933, −2.51821699423883497461446994450, −1.37388572967766001788558170868, 0,
1.37388572967766001788558170868, 2.51821699423883497461446994450, 3.79773994986688104433043088933, 5.41830921521966014793595626320, 5.97736110289531596964592855397, 6.83289301392679131193539115649, 7.56732007923316637212302928197, 9.086107904753971079322501010284, 9.541213078910765803227280733028
