| L(s) = 1 | − 1.15·2-s − 8.73·3-s − 6.67·4-s + 5·5-s + 10.0·6-s + 10.3·7-s + 16.9·8-s + 49.2·9-s − 5.76·10-s − 11·11-s + 58.2·12-s − 78.8·13-s − 11.9·14-s − 43.6·15-s + 33.8·16-s − 112.·17-s − 56.7·18-s − 19·19-s − 33.3·20-s − 90.2·21-s + 12.6·22-s + 177.·23-s − 147.·24-s + 25·25-s + 90.8·26-s − 194.·27-s − 68.9·28-s + ⋯ |
| L(s) = 1 | − 0.407·2-s − 1.68·3-s − 0.833·4-s + 0.447·5-s + 0.684·6-s + 0.558·7-s + 0.747·8-s + 1.82·9-s − 0.182·10-s − 0.301·11-s + 1.40·12-s − 1.68·13-s − 0.227·14-s − 0.751·15-s + 0.529·16-s − 1.60·17-s − 0.743·18-s − 0.229·19-s − 0.372·20-s − 0.938·21-s + 0.122·22-s + 1.60·23-s − 1.25·24-s + 0.200·25-s + 0.685·26-s − 1.38·27-s − 0.465·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 - 5T \) |
| 11 | \( 1 + 11T \) |
| 19 | \( 1 + 19T \) |
| good | 2 | \( 1 + 1.15T + 8T^{2} \) |
| 3 | \( 1 + 8.73T + 27T^{2} \) |
| 7 | \( 1 - 10.3T + 343T^{2} \) |
| 13 | \( 1 + 78.8T + 2.19e3T^{2} \) |
| 17 | \( 1 + 112.T + 4.91e3T^{2} \) |
| 23 | \( 1 - 177.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 29.0T + 2.43e4T^{2} \) |
| 31 | \( 1 - 75.1T + 2.97e4T^{2} \) |
| 37 | \( 1 - 34.9T + 5.06e4T^{2} \) |
| 41 | \( 1 - 194.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 239.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 147.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 627.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 40.4T + 2.05e5T^{2} \) |
| 61 | \( 1 - 799.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 42.9T + 3.00e5T^{2} \) |
| 71 | \( 1 - 226.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 518.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 532.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 57.3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 755.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 174.T + 9.12e5T^{2} \) |
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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
−9.328313390694531080294650132956, −8.385246014233620811325547985816, −7.24497647392228380271304440909, −6.64623394379261552897424259522, −5.38442985084519178740953434991, −4.95347598444922831136529501786, −4.33278268808930593418889217848, −2.31437413540624186171519838927, −0.936694566152370150183477865150, 0,
0.936694566152370150183477865150, 2.31437413540624186171519838927, 4.33278268808930593418889217848, 4.95347598444922831136529501786, 5.38442985084519178740953434991, 6.64623394379261552897424259522, 7.24497647392228380271304440909, 8.385246014233620811325547985816, 9.328313390694531080294650132956
