| L(s) = 1 | + 2·2-s + 3·3-s + 4·4-s − 12.9·5-s + 6·6-s − 25.6·7-s + 8·8-s + 9·9-s − 25.9·10-s + 26.7·11-s + 12·12-s + 8.59·13-s − 51.2·14-s − 38.9·15-s + 16·16-s + 8.16·17-s + 18·18-s − 51.9·20-s − 76.8·21-s + 53.5·22-s + 156.·23-s + 24·24-s + 43.6·25-s + 17.1·26-s + 27·27-s − 102.·28-s + 207.·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s − 1.16·5-s + 0.408·6-s − 1.38·7-s + 0.353·8-s + 0.333·9-s − 0.821·10-s + 0.733·11-s + 0.288·12-s + 0.183·13-s − 0.977·14-s − 0.670·15-s + 0.250·16-s + 0.116·17-s + 0.235·18-s − 0.580·20-s − 0.798·21-s + 0.518·22-s + 1.42·23-s + 0.204·24-s + 0.349·25-s + 0.129·26-s + 0.192·27-s − 0.691·28-s + 1.32·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2166 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2166 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - 2T \) |
| 3 | \( 1 - 3T \) |
| 19 | \( 1 \) |
| good | 5 | \( 1 + 12.9T + 125T^{2} \) |
| 7 | \( 1 + 25.6T + 343T^{2} \) |
| 11 | \( 1 - 26.7T + 1.33e3T^{2} \) |
| 13 | \( 1 - 8.59T + 2.19e3T^{2} \) |
| 17 | \( 1 - 8.16T + 4.91e3T^{2} \) |
| 23 | \( 1 - 156.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 207.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 94.5T + 2.97e4T^{2} \) |
| 37 | \( 1 + 197.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 376.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 508.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 366.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 203.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 592.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 509.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 250.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 115.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 832.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 369.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.28e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 61.0T + 7.04e5T^{2} \) |
| 97 | \( 1 + 247.T + 9.12e5T^{2} \) |
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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
−8.329107173473756270167649508272, −7.34919324829950716935968061536, −6.81903378770549736575854337293, −6.11775613268192312611817653965, −4.88803506370909219650598583290, −4.05811785024509853956879556387, −3.33748458512727510122371709184, −2.88802149649476282285472712290, −1.32274174276966197038975451196, 0,
1.32274174276966197038975451196, 2.88802149649476282285472712290, 3.33748458512727510122371709184, 4.05811785024509853956879556387, 4.88803506370909219650598583290, 6.11775613268192312611817653965, 6.81903378770549736575854337293, 7.34919324829950716935968061536, 8.329107173473756270167649508272
