| L(s) = 1 | − 2·2-s + 4·4-s + 18.4·5-s + 34.7·7-s − 8·8-s − 36.8·10-s + 29.1·11-s − 12.2·13-s − 69.5·14-s + 16·16-s + 12.2·17-s + 77.7·19-s + 73.6·20-s − 58.2·22-s − 6.32·23-s + 213.·25-s + 24.4·26-s + 139.·28-s − 133.·29-s − 151.·31-s − 32·32-s − 24.4·34-s + 640.·35-s − 162.·37-s − 155.·38-s − 147.·40-s + 341.·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s + 1.64·5-s + 1.87·7-s − 0.353·8-s − 1.16·10-s + 0.798·11-s − 0.260·13-s − 1.32·14-s + 0.250·16-s + 0.174·17-s + 0.938·19-s + 0.823·20-s − 0.564·22-s − 0.0573·23-s + 1.71·25-s + 0.184·26-s + 0.938·28-s − 0.857·29-s − 0.875·31-s − 0.176·32-s − 0.123·34-s + 3.09·35-s − 0.720·37-s − 0.663·38-s − 0.582·40-s + 1.30·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1458 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1458 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.226420643\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.226420643\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 2T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 18.4T + 125T^{2} \) |
| 7 | \( 1 - 34.7T + 343T^{2} \) |
| 11 | \( 1 - 29.1T + 1.33e3T^{2} \) |
| 13 | \( 1 + 12.2T + 2.19e3T^{2} \) |
| 17 | \( 1 - 12.2T + 4.91e3T^{2} \) |
| 19 | \( 1 - 77.7T + 6.85e3T^{2} \) |
| 23 | \( 1 + 6.32T + 1.21e4T^{2} \) |
| 29 | \( 1 + 133.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 151.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 162.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 341.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 347.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 151.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 329.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 229.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 719.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 385.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 125.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 369.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.00e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.13e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 650.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 720.T + 9.12e5T^{2} \) |
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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
−9.236360630001792221012010580876, −8.502632018993229428879035128190, −7.58571408356975146549079268208, −6.88270765071409552768971373646, −5.63453695833527094537289296256, −5.38369067580926386612164571248, −4.12055253630729154428865553855, −2.53240298264842650501625187245, −1.70760513474148989952168437494, −1.12342295885982334604331834512,
1.12342295885982334604331834512, 1.70760513474148989952168437494, 2.53240298264842650501625187245, 4.12055253630729154428865553855, 5.38369067580926386612164571248, 5.63453695833527094537289296256, 6.88270765071409552768971373646, 7.58571408356975146549079268208, 8.502632018993229428879035128190, 9.236360630001792221012010580876
