| L(s) = 1 | − 2-s − 3-s + 4-s + 6-s − 3.37·7-s − 8-s + 9-s + 0.627·11-s − 12-s + 2·13-s + 3.37·14-s + 16-s + 4.74·17-s − 18-s − 0.627·19-s + 3.37·21-s − 0.627·22-s − 3.37·23-s + 24-s − 2·26-s − 27-s − 3.37·28-s − 8.74·29-s − 31-s − 32-s − 0.627·33-s − 4.74·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.408·6-s − 1.27·7-s − 0.353·8-s + 0.333·9-s + 0.189·11-s − 0.288·12-s + 0.554·13-s + 0.901·14-s + 0.250·16-s + 1.15·17-s − 0.235·18-s − 0.144·19-s + 0.735·21-s − 0.133·22-s − 0.703·23-s + 0.204·24-s − 0.392·26-s − 0.192·27-s − 0.637·28-s − 1.62·29-s − 0.179·31-s − 0.176·32-s − 0.109·33-s − 0.813·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 31 | \( 1 + T \) |
| good | 7 | \( 1 + 3.37T + 7T^{2} \) |
| 11 | \( 1 - 0.627T + 11T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 17 | \( 1 - 4.74T + 17T^{2} \) |
| 19 | \( 1 + 0.627T + 19T^{2} \) |
| 23 | \( 1 + 3.37T + 23T^{2} \) |
| 29 | \( 1 + 8.74T + 29T^{2} \) |
| 37 | \( 1 - 0.744T + 37T^{2} \) |
| 41 | \( 1 - 0.744T + 41T^{2} \) |
| 43 | \( 1 + 0.627T + 43T^{2} \) |
| 47 | \( 1 - 6.74T + 47T^{2} \) |
| 53 | \( 1 + 1.37T + 53T^{2} \) |
| 59 | \( 1 - 2.74T + 59T^{2} \) |
| 61 | \( 1 - 11.4T + 61T^{2} \) |
| 67 | \( 1 + 10.7T + 67T^{2} \) |
| 71 | \( 1 - 3.37T + 71T^{2} \) |
| 73 | \( 1 - 8.11T + 73T^{2} \) |
| 79 | \( 1 + 4.62T + 79T^{2} \) |
| 83 | \( 1 - 12T + 83T^{2} \) |
| 89 | \( 1 + 1.37T + 89T^{2} \) |
| 97 | \( 1 + 2T + 97T^{2} \) |
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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
−7.84418435609534387069938295038, −7.28076407563719246707254811944, −6.44554811274053435676553966938, −5.96301252568829614000538049627, −5.28458938930947918254930968881, −3.91762770278380212533946480219, −3.42504671715502040127430119384, −2.26566659218720602915842411752, −1.10700420719223053737421382066, 0,
1.10700420719223053737421382066, 2.26566659218720602915842411752, 3.42504671715502040127430119384, 3.91762770278380212533946480219, 5.28458938930947918254930968881, 5.96301252568829614000538049627, 6.44554811274053435676553966938, 7.28076407563719246707254811944, 7.84418435609534387069938295038
