| L(s) = 1 | + 3.56·2-s + 4.68·4-s + 7·7-s − 11.8·8-s + 5.19·11-s − 54.5·13-s + 24.9·14-s − 79.5·16-s + 16.1·17-s + 87.4·19-s + 18.4·22-s + 176.·23-s − 194.·26-s + 32.7·28-s − 142.·29-s − 94.3·31-s − 188.·32-s + 57.5·34-s − 17.3·37-s + 311.·38-s − 210.·41-s − 521.·43-s + 24.3·44-s + 628.·46-s − 105.·47-s + 49·49-s − 255.·52-s + ⋯ |
| L(s) = 1 | + 1.25·2-s + 0.585·4-s + 0.377·7-s − 0.521·8-s + 0.142·11-s − 1.16·13-s + 0.475·14-s − 1.24·16-s + 0.230·17-s + 1.05·19-s + 0.179·22-s + 1.59·23-s − 1.46·26-s + 0.221·28-s − 0.910·29-s − 0.546·31-s − 1.04·32-s + 0.290·34-s − 0.0770·37-s + 1.32·38-s − 0.800·41-s − 1.84·43-s + 0.0833·44-s + 2.01·46-s − 0.327·47-s + 0.142·49-s − 0.681·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - 7T \) |
| good | 2 | \( 1 - 3.56T + 8T^{2} \) |
| 11 | \( 1 - 5.19T + 1.33e3T^{2} \) |
| 13 | \( 1 + 54.5T + 2.19e3T^{2} \) |
| 17 | \( 1 - 16.1T + 4.91e3T^{2} \) |
| 19 | \( 1 - 87.4T + 6.85e3T^{2} \) |
| 23 | \( 1 - 176.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 142.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 94.3T + 2.97e4T^{2} \) |
| 37 | \( 1 + 17.3T + 5.06e4T^{2} \) |
| 41 | \( 1 + 210.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 521.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 105.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 108.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 210.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 674.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 324.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 793.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 315.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 425.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 283.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 843.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.53e3T + 9.12e5T^{2} \) |
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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
−8.765312914968619364598855332762, −7.55548946289411703870080051325, −7.00040822047829722643853033199, −5.94215564109347423173408692149, −5.06437060952825914625600459184, −4.74839546286551991652220156230, −3.50723518173021526010334787445, −2.87729792334582457553884468573, −1.59212372667214328464560684939, 0,
1.59212372667214328464560684939, 2.87729792334582457553884468573, 3.50723518173021526010334787445, 4.74839546286551991652220156230, 5.06437060952825914625600459184, 5.94215564109347423173408692149, 7.00040822047829722643853033199, 7.55548946289411703870080051325, 8.765312914968619364598855332762
