| L(s) = 1 | + 2-s + 4-s − 4·7-s + 8-s − 3·11-s + 2·13-s − 4·14-s + 16-s + 6·17-s − 3·22-s + 6·23-s − 5·25-s + 2·26-s − 4·28-s + 2·31-s + 32-s + 6·34-s − 10·37-s − 9·41-s − 4·43-s − 3·44-s + 6·46-s + 9·49-s − 5·50-s + 2·52-s − 6·53-s − 4·56-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 1.51·7-s + 0.353·8-s − 0.904·11-s + 0.554·13-s − 1.06·14-s + 1/4·16-s + 1.45·17-s − 0.639·22-s + 1.25·23-s − 25-s + 0.392·26-s − 0.755·28-s + 0.359·31-s + 0.176·32-s + 1.02·34-s − 1.64·37-s − 1.40·41-s − 0.609·43-s − 0.452·44-s + 0.884·46-s + 9/7·49-s − 0.707·50-s + 0.277·52-s − 0.824·53-s − 0.534·56-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6498 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6498 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 3 | \( 1 \) |
| 19 | \( 1 \) |
| good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 9 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 + 7 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 3 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 17 T + p T^{2} \) |
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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
−7.48769421530084903056992246264, −6.82597538078964472544369886283, −6.20806267993305578119789112794, −5.47562787816053701111108078834, −4.97298216866026274237899856437, −3.70475296867170937340934834002, −3.34397809677612158634689897147, −2.64886271651021039964255402082, −1.39793116430671782406980466100, 0,
1.39793116430671782406980466100, 2.64886271651021039964255402082, 3.34397809677612158634689897147, 3.70475296867170937340934834002, 4.97298216866026274237899856437, 5.47562787816053701111108078834, 6.20806267993305578119789112794, 6.82597538078964472544369886283, 7.48769421530084903056992246264
