| L(s) = 1 | + 0.886·2-s + 3-s − 1.21·4-s − 5-s + 0.886·6-s + 2.28·7-s − 2.84·8-s + 9-s − 0.886·10-s + 2.96·11-s − 1.21·12-s + 0.0808·13-s + 2.02·14-s − 15-s − 0.100·16-s − 6.24·17-s + 0.886·18-s − 5.30·19-s + 1.21·20-s + 2.28·21-s + 2.62·22-s + 7.96·23-s − 2.84·24-s + 25-s + 0.0717·26-s + 27-s − 2.76·28-s + ⋯ |
| L(s) = 1 | + 0.627·2-s + 0.577·3-s − 0.606·4-s − 0.447·5-s + 0.362·6-s + 0.862·7-s − 1.00·8-s + 0.333·9-s − 0.280·10-s + 0.893·11-s − 0.350·12-s + 0.0224·13-s + 0.541·14-s − 0.258·15-s − 0.0252·16-s − 1.51·17-s + 0.209·18-s − 1.21·19-s + 0.271·20-s + 0.498·21-s + 0.560·22-s + 1.66·23-s − 0.581·24-s + 0.200·25-s + 0.0140·26-s + 0.192·27-s − 0.523·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 401 | \( 1 - T \) |
| good | 2 | \( 1 - 0.886T + 2T^{2} \) |
| 7 | \( 1 - 2.28T + 7T^{2} \) |
| 11 | \( 1 - 2.96T + 11T^{2} \) |
| 13 | \( 1 - 0.0808T + 13T^{2} \) |
| 17 | \( 1 + 6.24T + 17T^{2} \) |
| 19 | \( 1 + 5.30T + 19T^{2} \) |
| 23 | \( 1 - 7.96T + 23T^{2} \) |
| 29 | \( 1 + 6.23T + 29T^{2} \) |
| 31 | \( 1 + 8.56T + 31T^{2} \) |
| 37 | \( 1 - 0.640T + 37T^{2} \) |
| 41 | \( 1 + 9.60T + 41T^{2} \) |
| 43 | \( 1 + 0.186T + 43T^{2} \) |
| 47 | \( 1 + 1.46T + 47T^{2} \) |
| 53 | \( 1 - 0.120T + 53T^{2} \) |
| 59 | \( 1 + 4.08T + 59T^{2} \) |
| 61 | \( 1 - 1.96T + 61T^{2} \) |
| 67 | \( 1 - 8.29T + 67T^{2} \) |
| 71 | \( 1 - 4.32T + 71T^{2} \) |
| 73 | \( 1 - 11.8T + 73T^{2} \) |
| 79 | \( 1 + 16.8T + 79T^{2} \) |
| 83 | \( 1 - 4.32T + 83T^{2} \) |
| 89 | \( 1 - 8.91T + 89T^{2} \) |
| 97 | \( 1 + 2.89T + 97T^{2} \) |
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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
−7.81991197559407208345466665204, −6.91620255286716374328425038743, −6.40891951077203750387776470168, −5.22034170533959125682500452900, −4.77822765705475875910542147583, −3.96774180395104739956293974490, −3.58572731444487707174354607275, −2.42429825959154378060574749903, −1.49582375078790161767227897711, 0,
1.49582375078790161767227897711, 2.42429825959154378060574749903, 3.58572731444487707174354607275, 3.96774180395104739956293974490, 4.77822765705475875910542147583, 5.22034170533959125682500452900, 6.40891951077203750387776470168, 6.91620255286716374328425038743, 7.81991197559407208345466665204
