| L(s) = 1 | + 0.577·2-s + 0.746·3-s − 1.66·4-s + 0.431·6-s + 4.19·7-s − 2.11·8-s − 2.44·9-s − 11-s − 1.24·12-s − 0.324·13-s + 2.42·14-s + 2.10·16-s + 5.00·17-s − 1.41·18-s − 19-s + 3.13·21-s − 0.577·22-s − 8.11·23-s − 1.58·24-s − 0.187·26-s − 4.06·27-s − 6.98·28-s − 3.53·29-s − 3.45·31-s + 5.45·32-s − 0.746·33-s + 2.89·34-s + ⋯ |
| L(s) = 1 | + 0.408·2-s + 0.431·3-s − 0.833·4-s + 0.176·6-s + 1.58·7-s − 0.748·8-s − 0.814·9-s − 0.301·11-s − 0.359·12-s − 0.0899·13-s + 0.647·14-s + 0.527·16-s + 1.21·17-s − 0.332·18-s − 0.229·19-s + 0.683·21-s − 0.123·22-s − 1.69·23-s − 0.322·24-s − 0.0367·26-s − 0.782·27-s − 1.32·28-s − 0.656·29-s − 0.621·31-s + 0.964·32-s − 0.129·33-s + 0.496·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| good | 2 | \( 1 - 0.577T + 2T^{2} \) |
| 3 | \( 1 - 0.746T + 3T^{2} \) |
| 7 | \( 1 - 4.19T + 7T^{2} \) |
| 13 | \( 1 + 0.324T + 13T^{2} \) |
| 17 | \( 1 - 5.00T + 17T^{2} \) |
| 23 | \( 1 + 8.11T + 23T^{2} \) |
| 29 | \( 1 + 3.53T + 29T^{2} \) |
| 31 | \( 1 + 3.45T + 31T^{2} \) |
| 37 | \( 1 + 5.77T + 37T^{2} \) |
| 41 | \( 1 - 0.484T + 41T^{2} \) |
| 43 | \( 1 + 5.86T + 43T^{2} \) |
| 47 | \( 1 + 6.22T + 47T^{2} \) |
| 53 | \( 1 - 10.2T + 53T^{2} \) |
| 59 | \( 1 - 7.16T + 59T^{2} \) |
| 61 | \( 1 - 5.33T + 61T^{2} \) |
| 67 | \( 1 + 8.20T + 67T^{2} \) |
| 71 | \( 1 + 10.0T + 71T^{2} \) |
| 73 | \( 1 + 2.88T + 73T^{2} \) |
| 79 | \( 1 - 3.95T + 79T^{2} \) |
| 83 | \( 1 + 10.2T + 83T^{2} \) |
| 89 | \( 1 - 7.47T + 89T^{2} \) |
| 97 | \( 1 - 13.9T + 97T^{2} \) |
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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
−7.946631232383100846224297559307, −7.47334149668509205540520263524, −6.10089062239668782283651762870, −5.40992697541744812891551376744, −5.06487367925202319678820731882, −4.05099374802824709313475731003, −3.50751269787072095913438281317, −2.41824787307215002851607943679, −1.49417526420053983440673895278, 0,
1.49417526420053983440673895278, 2.41824787307215002851607943679, 3.50751269787072095913438281317, 4.05099374802824709313475731003, 5.06487367925202319678820731882, 5.40992697541744812891551376744, 6.10089062239668782283651762870, 7.47334149668509205540520263524, 7.946631232383100846224297559307
