| L(s) = 1 | + 2-s + 4-s + 2.26·5-s − 0.397·7-s + 8-s + 2.26·10-s − 3.96·11-s + 0.823·13-s − 0.397·14-s + 16-s + 6.85·17-s − 8.38·19-s + 2.26·20-s − 3.96·22-s − 8.28·23-s + 0.142·25-s + 0.823·26-s − 0.397·28-s − 8.78·29-s + 4.00·31-s + 32-s + 6.85·34-s − 0.902·35-s + 0.694·37-s − 8.38·38-s + 2.26·40-s − 9.11·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s + 1.01·5-s − 0.150·7-s + 0.353·8-s + 0.717·10-s − 1.19·11-s + 0.228·13-s − 0.106·14-s + 0.250·16-s + 1.66·17-s − 1.92·19-s + 0.507·20-s − 0.846·22-s − 1.72·23-s + 0.0284·25-s + 0.161·26-s − 0.0751·28-s − 1.63·29-s + 0.719·31-s + 0.176·32-s + 1.17·34-s − 0.152·35-s + 0.114·37-s − 1.36·38-s + 0.358·40-s − 1.42·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8046 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8046 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 149 | \( 1 + T \) |
| good | 5 | \( 1 - 2.26T + 5T^{2} \) |
| 7 | \( 1 + 0.397T + 7T^{2} \) |
| 11 | \( 1 + 3.96T + 11T^{2} \) |
| 13 | \( 1 - 0.823T + 13T^{2} \) |
| 17 | \( 1 - 6.85T + 17T^{2} \) |
| 19 | \( 1 + 8.38T + 19T^{2} \) |
| 23 | \( 1 + 8.28T + 23T^{2} \) |
| 29 | \( 1 + 8.78T + 29T^{2} \) |
| 31 | \( 1 - 4.00T + 31T^{2} \) |
| 37 | \( 1 - 0.694T + 37T^{2} \) |
| 41 | \( 1 + 9.11T + 41T^{2} \) |
| 43 | \( 1 + 8.24T + 43T^{2} \) |
| 47 | \( 1 + 1.58T + 47T^{2} \) |
| 53 | \( 1 - 7.80T + 53T^{2} \) |
| 59 | \( 1 - 10.9T + 59T^{2} \) |
| 61 | \( 1 + 8.40T + 61T^{2} \) |
| 67 | \( 1 + 11.4T + 67T^{2} \) |
| 71 | \( 1 + 14.7T + 71T^{2} \) |
| 73 | \( 1 - 6.27T + 73T^{2} \) |
| 79 | \( 1 - 9.64T + 79T^{2} \) |
| 83 | \( 1 + 8.76T + 83T^{2} \) |
| 89 | \( 1 - 9.24T + 89T^{2} \) |
| 97 | \( 1 + 8.05T + 97T^{2} \) |
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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
−7.49612392999906056343875498073, −6.49308898794928812888309787313, −5.96335468946767199708779446679, −5.52377474914308804424634542866, −4.78119430802743334003368631943, −3.86837097678792025794666365368, −3.15203521698244114767425475448, −2.16896515756365228113377964976, −1.71242002893776882829214890649, 0,
1.71242002893776882829214890649, 2.16896515756365228113377964976, 3.15203521698244114767425475448, 3.86837097678792025794666365368, 4.78119430802743334003368631943, 5.52377474914308804424634542866, 5.96335468946767199708779446679, 6.49308898794928812888309787313, 7.49612392999906056343875498073
