| L(s) = 1 | + 1.22·2-s − 6.49·4-s + 2.21·5-s + 7·7-s − 17.7·8-s + 2.71·10-s − 11·11-s + 12.8·13-s + 8.59·14-s + 30.0·16-s + 45.5·17-s + 11.0·19-s − 14.3·20-s − 13.5·22-s − 177.·23-s − 120.·25-s + 15.7·26-s − 45.4·28-s − 58.5·29-s + 175.·31-s + 179.·32-s + 55.9·34-s + 15.4·35-s + 221.·37-s + 13.5·38-s − 39.3·40-s + 307.·41-s + ⋯ |
| L(s) = 1 | + 0.434·2-s − 0.811·4-s + 0.197·5-s + 0.377·7-s − 0.786·8-s + 0.0858·10-s − 0.301·11-s + 0.273·13-s + 0.164·14-s + 0.470·16-s + 0.649·17-s + 0.133·19-s − 0.160·20-s − 0.130·22-s − 1.60·23-s − 0.960·25-s + 0.118·26-s − 0.306·28-s − 0.374·29-s + 1.01·31-s + 0.990·32-s + 0.282·34-s + 0.0747·35-s + 0.982·37-s + 0.0579·38-s − 0.155·40-s + 1.17·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.964375833\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.964375833\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 - 7T \) |
| 11 | \( 1 + 11T \) |
| good | 2 | \( 1 - 1.22T + 8T^{2} \) |
| 5 | \( 1 - 2.21T + 125T^{2} \) |
| 13 | \( 1 - 12.8T + 2.19e3T^{2} \) |
| 17 | \( 1 - 45.5T + 4.91e3T^{2} \) |
| 19 | \( 1 - 11.0T + 6.85e3T^{2} \) |
| 23 | \( 1 + 177.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 58.5T + 2.43e4T^{2} \) |
| 31 | \( 1 - 175.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 221.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 307.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 462.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 293.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 400.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 16.3T + 2.05e5T^{2} \) |
| 61 | \( 1 - 509.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 483.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 202.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 885.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 289.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 106.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.58e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 990.T + 9.12e5T^{2} \) |
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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
−9.895279332279859027555959665786, −9.358748374935374598237579158646, −8.206300688498899765657519676710, −7.70497016751104457066541371631, −6.12798933732027149468035941235, −5.59203674547678554471394167824, −4.47536132503361889162416236550, −3.71697840627630765893676486736, −2.35793465870745823918766976541, −0.76870087383941852693374709997,
0.76870087383941852693374709997, 2.35793465870745823918766976541, 3.71697840627630765893676486736, 4.47536132503361889162416236550, 5.59203674547678554471394167824, 6.12798933732027149468035941235, 7.70497016751104457066541371631, 8.206300688498899765657519676710, 9.358748374935374598237579158646, 9.895279332279859027555959665786
