| L(s) = 1 | + 0.787·3-s + 5-s + 1.37·7-s − 2.37·9-s − 6.04·11-s + 3.25·13-s + 0.787·15-s − 4.23·17-s − 19-s + 1.08·21-s − 0.0553·23-s + 25-s − 4.23·27-s + 2.66·29-s − 0.816·31-s − 4.75·33-s + 1.37·35-s − 6.30·37-s + 2.56·39-s + 1.57·41-s − 0.717·43-s − 2.37·45-s − 7.22·47-s − 5.09·49-s − 3.33·51-s − 10.7·53-s − 6.04·55-s + ⋯ |
| L(s) = 1 | + 0.454·3-s + 0.447·5-s + 0.521·7-s − 0.793·9-s − 1.82·11-s + 0.902·13-s + 0.203·15-s − 1.02·17-s − 0.229·19-s + 0.237·21-s − 0.0115·23-s + 0.200·25-s − 0.815·27-s + 0.494·29-s − 0.146·31-s − 0.828·33-s + 0.233·35-s − 1.03·37-s + 0.410·39-s + 0.246·41-s − 0.109·43-s − 0.354·45-s − 1.05·47-s − 0.728·49-s − 0.467·51-s − 1.47·53-s − 0.814·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - 0.787T + 3T^{2} \) |
| 7 | \( 1 - 1.37T + 7T^{2} \) |
| 11 | \( 1 + 6.04T + 11T^{2} \) |
| 13 | \( 1 - 3.25T + 13T^{2} \) |
| 17 | \( 1 + 4.23T + 17T^{2} \) |
| 23 | \( 1 + 0.0553T + 23T^{2} \) |
| 29 | \( 1 - 2.66T + 29T^{2} \) |
| 31 | \( 1 + 0.816T + 31T^{2} \) |
| 37 | \( 1 + 6.30T + 37T^{2} \) |
| 41 | \( 1 - 1.57T + 41T^{2} \) |
| 43 | \( 1 + 0.717T + 43T^{2} \) |
| 47 | \( 1 + 7.22T + 47T^{2} \) |
| 53 | \( 1 + 10.7T + 53T^{2} \) |
| 59 | \( 1 - 3.84T + 59T^{2} \) |
| 61 | \( 1 - 8.66T + 61T^{2} \) |
| 67 | \( 1 + 5.40T + 67T^{2} \) |
| 71 | \( 1 + 12.5T + 71T^{2} \) |
| 73 | \( 1 - 4.48T + 73T^{2} \) |
| 79 | \( 1 + 7.43T + 79T^{2} \) |
| 83 | \( 1 + 3.28T + 83T^{2} \) |
| 89 | \( 1 + 1.74T + 89T^{2} \) |
| 97 | \( 1 - 4.59T + 97T^{2} \) |
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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
−8.363179111820342379945455711026, −7.86537285569929029977528740274, −6.83746982998920667059769211347, −5.97070409071839289585331723101, −5.27743909470264034840553499993, −4.55462632570110411004078965645, −3.32410547714034747014805317124, −2.58676832003647121027052453579, −1.73869557840119270028115244618, 0,
1.73869557840119270028115244618, 2.58676832003647121027052453579, 3.32410547714034747014805317124, 4.55462632570110411004078965645, 5.27743909470264034840553499993, 5.97070409071839289585331723101, 6.83746982998920667059769211347, 7.86537285569929029977528740274, 8.363179111820342379945455711026
