| L(s) = 1 | − 1.76·3-s + 5-s + 3.89·7-s + 0.114·9-s + 4.80·11-s − 0.0353·13-s − 1.76·15-s − 4.39·17-s + 19-s − 6.86·21-s − 9.12·23-s + 25-s + 5.09·27-s + 6.21·29-s − 9.71·31-s − 8.48·33-s + 3.89·35-s + 0.749·37-s + 0.0623·39-s − 3.52·41-s − 11.4·43-s + 0.114·45-s − 5.57·47-s + 8.15·49-s + 7.75·51-s − 8.67·53-s + 4.80·55-s + ⋯ |
| L(s) = 1 | − 1.01·3-s + 0.447·5-s + 1.47·7-s + 0.0382·9-s + 1.44·11-s − 0.00979·13-s − 0.455·15-s − 1.06·17-s + 0.229·19-s − 1.49·21-s − 1.90·23-s + 0.200·25-s + 0.979·27-s + 1.15·29-s − 1.74·31-s − 1.47·33-s + 0.657·35-s + 0.123·37-s + 0.00997·39-s − 0.551·41-s − 1.74·43-s + 0.0171·45-s − 0.813·47-s + 1.16·49-s + 1.08·51-s − 1.19·53-s + 0.648·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 + 1.76T + 3T^{2} \) |
| 7 | \( 1 - 3.89T + 7T^{2} \) |
| 11 | \( 1 - 4.80T + 11T^{2} \) |
| 13 | \( 1 + 0.0353T + 13T^{2} \) |
| 17 | \( 1 + 4.39T + 17T^{2} \) |
| 23 | \( 1 + 9.12T + 23T^{2} \) |
| 29 | \( 1 - 6.21T + 29T^{2} \) |
| 31 | \( 1 + 9.71T + 31T^{2} \) |
| 37 | \( 1 - 0.749T + 37T^{2} \) |
| 41 | \( 1 + 3.52T + 41T^{2} \) |
| 43 | \( 1 + 11.4T + 43T^{2} \) |
| 47 | \( 1 + 5.57T + 47T^{2} \) |
| 53 | \( 1 + 8.67T + 53T^{2} \) |
| 59 | \( 1 + 2.56T + 59T^{2} \) |
| 61 | \( 1 + 6.50T + 61T^{2} \) |
| 67 | \( 1 + 13.2T + 67T^{2} \) |
| 71 | \( 1 + 1.67T + 71T^{2} \) |
| 73 | \( 1 + 11.3T + 73T^{2} \) |
| 79 | \( 1 + 11.4T + 79T^{2} \) |
| 83 | \( 1 + 5.64T + 83T^{2} \) |
| 89 | \( 1 + 1.24T + 89T^{2} \) |
| 97 | \( 1 - 9.31T + 97T^{2} \) |
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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
−7.67473150371267126976698998785, −6.81556778115561988107827428901, −6.19059697913385243541765483513, −5.69102075392839978932891434612, −4.70291604910052494802852613930, −4.46682178804743134727003378707, −3.29981202419280154625228770197, −1.87986725568494665260665604895, −1.48504426948276901845319938933, 0,
1.48504426948276901845319938933, 1.87986725568494665260665604895, 3.29981202419280154625228770197, 4.46682178804743134727003378707, 4.70291604910052494802852613930, 5.69102075392839978932891434612, 6.19059697913385243541765483513, 6.81556778115561988107827428901, 7.67473150371267126976698998785
