| L(s) = 1 | − 0.667·2-s + 2.21·3-s − 1.55·4-s − 1.47·6-s − 0.972·7-s + 2.37·8-s + 1.89·9-s + 1.85·11-s − 3.43·12-s − 3.42·13-s + 0.649·14-s + 1.52·16-s − 2.34·17-s − 1.26·18-s + 1.56·19-s − 2.15·21-s − 1.23·22-s + 4.42·23-s + 5.24·24-s + 2.28·26-s − 2.44·27-s + 1.51·28-s + 6.40·29-s − 7.34·31-s − 5.76·32-s + 4.10·33-s + 1.56·34-s + ⋯ |
| L(s) = 1 | − 0.472·2-s + 1.27·3-s − 0.777·4-s − 0.603·6-s − 0.367·7-s + 0.839·8-s + 0.631·9-s + 0.559·11-s − 0.992·12-s − 0.949·13-s + 0.173·14-s + 0.380·16-s − 0.569·17-s − 0.298·18-s + 0.360·19-s − 0.469·21-s − 0.264·22-s + 0.923·23-s + 1.07·24-s + 0.448·26-s − 0.470·27-s + 0.285·28-s + 1.19·29-s − 1.31·31-s − 1.01·32-s + 0.714·33-s + 0.268·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 \) |
| 241 | \( 1 + T \) |
| good | 2 | \( 1 + 0.667T + 2T^{2} \) |
| 3 | \( 1 - 2.21T + 3T^{2} \) |
| 7 | \( 1 + 0.972T + 7T^{2} \) |
| 11 | \( 1 - 1.85T + 11T^{2} \) |
| 13 | \( 1 + 3.42T + 13T^{2} \) |
| 17 | \( 1 + 2.34T + 17T^{2} \) |
| 19 | \( 1 - 1.56T + 19T^{2} \) |
| 23 | \( 1 - 4.42T + 23T^{2} \) |
| 29 | \( 1 - 6.40T + 29T^{2} \) |
| 31 | \( 1 + 7.34T + 31T^{2} \) |
| 37 | \( 1 + 1.11T + 37T^{2} \) |
| 41 | \( 1 + 4.63T + 41T^{2} \) |
| 43 | \( 1 + 9.19T + 43T^{2} \) |
| 47 | \( 1 - 3.29T + 47T^{2} \) |
| 53 | \( 1 - 2.62T + 53T^{2} \) |
| 59 | \( 1 - 5.88T + 59T^{2} \) |
| 61 | \( 1 - 0.295T + 61T^{2} \) |
| 67 | \( 1 + 7.96T + 67T^{2} \) |
| 71 | \( 1 + 0.815T + 71T^{2} \) |
| 73 | \( 1 - 13.3T + 73T^{2} \) |
| 79 | \( 1 + 5.59T + 79T^{2} \) |
| 83 | \( 1 - 11.2T + 83T^{2} \) |
| 89 | \( 1 - 1.69T + 89T^{2} \) |
| 97 | \( 1 + 4.82T + 97T^{2} \) |
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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
−7.966203854185510959420993905679, −7.18829215441486555867511911620, −6.64441542675895569719250646622, −5.37083966439596950312890716165, −4.75428398331881163635016749176, −3.85601703421023468206438773722, −3.24322316538674631605729448232, −2.35895989733628439022495758385, −1.36117897635540300723011323986, 0,
1.36117897635540300723011323986, 2.35895989733628439022495758385, 3.24322316538674631605729448232, 3.85601703421023468206438773722, 4.75428398331881163635016749176, 5.37083966439596950312890716165, 6.64441542675895569719250646622, 7.18829215441486555867511911620, 7.966203854185510959420993905679
