| L(s) = 1 | − 5·5-s − 19.6·7-s + 29.1·11-s − 13.6·13-s − 39.0·17-s + 43.7·19-s + 159.·23-s + 25·25-s − 25.1·29-s − 116.·31-s + 98.1·35-s + 329.·37-s + 180.·41-s − 213.·43-s − 179.·47-s + 42.4·49-s + 60.8·53-s − 145.·55-s − 382.·59-s + 433.·61-s + 68.1·65-s + 125.·67-s − 30.0·71-s + 676.·73-s − 572.·77-s − 833.·79-s − 353.·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 1.06·7-s + 0.799·11-s − 0.290·13-s − 0.557·17-s + 0.528·19-s + 1.44·23-s + 0.200·25-s − 0.161·29-s − 0.676·31-s + 0.474·35-s + 1.46·37-s + 0.688·41-s − 0.757·43-s − 0.556·47-s + 0.123·49-s + 0.157·53-s − 0.357·55-s − 0.844·59-s + 0.910·61-s + 0.130·65-s + 0.228·67-s − 0.0502·71-s + 1.08·73-s − 0.847·77-s − 1.18·79-s − 0.467·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + 5T \) |
| good | 7 | \( 1 + 19.6T + 343T^{2} \) |
| 11 | \( 1 - 29.1T + 1.33e3T^{2} \) |
| 13 | \( 1 + 13.6T + 2.19e3T^{2} \) |
| 17 | \( 1 + 39.0T + 4.91e3T^{2} \) |
| 19 | \( 1 - 43.7T + 6.85e3T^{2} \) |
| 23 | \( 1 - 159.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 25.1T + 2.43e4T^{2} \) |
| 31 | \( 1 + 116.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 329.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 180.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 213.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 179.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 60.8T + 1.48e5T^{2} \) |
| 59 | \( 1 + 382.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 433.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 125.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 30.0T + 3.57e5T^{2} \) |
| 73 | \( 1 - 676.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 833.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 353.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 948.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 285.T + 9.12e5T^{2} \) |
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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
−8.810945508707834938492069900602, −7.74662972488157448633493977661, −6.93729177726952350585945657068, −6.39667330769978101632280975753, −5.34058446483715289741403172676, −4.33456030899922310578026457219, −3.47457927272148315089367468525, −2.64837635614331669194645523081, −1.16948339386518633314267771451, 0,
1.16948339386518633314267771451, 2.64837635614331669194645523081, 3.47457927272148315089367468525, 4.33456030899922310578026457219, 5.34058446483715289741403172676, 6.39667330769978101632280975753, 6.93729177726952350585945657068, 7.74662972488157448633493977661, 8.810945508707834938492069900602
