| L(s) = 1 | − 2·2-s − 3·3-s + 4·4-s + 5·5-s + 6·6-s − 19·7-s − 8·8-s + 9·9-s − 10·10-s − 24·11-s − 12·12-s + 44·13-s + 38·14-s − 15·15-s + 16·16-s + 75·17-s − 18·18-s − 16·19-s + 20·20-s + 57·21-s + 48·22-s − 23·23-s + 24·24-s + 25·25-s − 88·26-s − 27·27-s − 76·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s − 1.02·7-s − 0.353·8-s + 1/3·9-s − 0.316·10-s − 0.657·11-s − 0.288·12-s + 0.938·13-s + 0.725·14-s − 0.258·15-s + 1/4·16-s + 1.07·17-s − 0.235·18-s − 0.193·19-s + 0.223·20-s + 0.592·21-s + 0.465·22-s − 0.208·23-s + 0.204·24-s + 1/5·25-s − 0.663·26-s − 0.192·27-s − 0.512·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + p T \) |
| 3 | \( 1 + p T \) |
| 5 | \( 1 - p T \) |
| 23 | \( 1 + p T \) |
| good | 7 | \( 1 + 19 T + p^{3} T^{2} \) |
| 11 | \( 1 + 24 T + p^{3} T^{2} \) |
| 13 | \( 1 - 44 T + p^{3} T^{2} \) |
| 17 | \( 1 - 75 T + p^{3} T^{2} \) |
| 19 | \( 1 + 16 T + p^{3} T^{2} \) |
| 29 | \( 1 + 123 T + p^{3} T^{2} \) |
| 31 | \( 1 + 43 T + p^{3} T^{2} \) |
| 37 | \( 1 + 43 T + p^{3} T^{2} \) |
| 41 | \( 1 - 207 T + p^{3} T^{2} \) |
| 43 | \( 1 - 236 T + p^{3} T^{2} \) |
| 47 | \( 1 + 30 T + p^{3} T^{2} \) |
| 53 | \( 1 - 519 T + p^{3} T^{2} \) |
| 59 | \( 1 - 39 T + p^{3} T^{2} \) |
| 61 | \( 1 + 190 T + p^{3} T^{2} \) |
| 67 | \( 1 + 295 T + p^{3} T^{2} \) |
| 71 | \( 1 + 603 T + p^{3} T^{2} \) |
| 73 | \( 1 - 668 T + p^{3} T^{2} \) |
| 79 | \( 1 + 1276 T + p^{3} T^{2} \) |
| 83 | \( 1 + 573 T + p^{3} T^{2} \) |
| 89 | \( 1 - 456 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1186 T + p^{3} T^{2} \) |
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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
−9.732284519936668521927741988885, −8.955862766182411448229849998272, −7.88511787675907276840478511041, −6.98594788784380040625703820608, −6.03300275501891073219286674952, −5.50539960391504948354731441913, −3.87497308908400701397697154013, −2.72620936090424641501176099382, −1.28058594126146580222636944374, 0,
1.28058594126146580222636944374, 2.72620936090424641501176099382, 3.87497308908400701397697154013, 5.50539960391504948354731441913, 6.03300275501891073219286674952, 6.98594788784380040625703820608, 7.88511787675907276840478511041, 8.955862766182411448229849998272, 9.732284519936668521927741988885
