| L(s) = 1 | + 2-s + 4-s + 3.53·5-s + 3.71·7-s + 8-s + 3.53·10-s + 5.29·11-s + 0.226·13-s + 3.71·14-s + 16-s + 1.65·17-s + 3.53·20-s + 5.29·22-s − 8.68·23-s + 7.47·25-s + 0.226·26-s + 3.71·28-s + 0.120·29-s − 3.12·31-s + 32-s + 1.65·34-s + 13.1·35-s − 5.12·37-s + 3.53·40-s + 7.10·41-s − 5.35·43-s + 5.29·44-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s + 1.57·5-s + 1.40·7-s + 0.353·8-s + 1.11·10-s + 1.59·11-s + 0.0628·13-s + 0.993·14-s + 0.250·16-s + 0.400·17-s + 0.789·20-s + 1.12·22-s − 1.80·23-s + 1.49·25-s + 0.0444·26-s + 0.702·28-s + 0.0223·29-s − 0.560·31-s + 0.176·32-s + 0.283·34-s + 2.21·35-s − 0.843·37-s + 0.558·40-s + 1.10·41-s − 0.816·43-s + 0.797·44-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
| \(L(1)\) |
\(\approx\) |
\(6.009746580\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.009746580\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 3.53T + 5T^{2} \) |
| 7 | \( 1 - 3.71T + 7T^{2} \) |
| 11 | \( 1 - 5.29T + 11T^{2} \) |
| 13 | \( 1 - 0.226T + 13T^{2} \) |
| 17 | \( 1 - 1.65T + 17T^{2} \) |
| 23 | \( 1 + 8.68T + 23T^{2} \) |
| 29 | \( 1 - 0.120T + 29T^{2} \) |
| 31 | \( 1 + 3.12T + 31T^{2} \) |
| 37 | \( 1 + 5.12T + 37T^{2} \) |
| 41 | \( 1 - 7.10T + 41T^{2} \) |
| 43 | \( 1 + 5.35T + 43T^{2} \) |
| 47 | \( 1 - 2.50T + 47T^{2} \) |
| 53 | \( 1 - 5.93T + 53T^{2} \) |
| 59 | \( 1 - 0.218T + 59T^{2} \) |
| 61 | \( 1 + 1.57T + 61T^{2} \) |
| 67 | \( 1 + 15.4T + 67T^{2} \) |
| 71 | \( 1 + 1.35T + 71T^{2} \) |
| 73 | \( 1 - 2.42T + 73T^{2} \) |
| 79 | \( 1 - 2.86T + 79T^{2} \) |
| 83 | \( 1 + 1.92T + 83T^{2} \) |
| 89 | \( 1 + 12.1T + 89T^{2} \) |
| 97 | \( 1 + 17.5T + 97T^{2} \) |
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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
−7.984661125427572754407893608687, −7.11769333217839168566585682161, −6.39541108456756797706873584534, −5.76814355139124554169361139434, −5.32844023700967230934182145900, −4.38557885211296306896540085694, −3.83909151720908378820089120183, −2.60769611221493560973979216801, −1.69602017877664285911382971487, −1.42916699282280002646744914352,
1.42916699282280002646744914352, 1.69602017877664285911382971487, 2.60769611221493560973979216801, 3.83909151720908378820089120183, 4.38557885211296306896540085694, 5.32844023700967230934182145900, 5.76814355139124554169361139434, 6.39541108456756797706873584534, 7.11769333217839168566585682161, 7.984661125427572754407893608687
