| L(s) = 1 | + 2·2-s − 2.10·3-s + 4·4-s − 4.21·6-s + 35.7·7-s + 8·8-s − 22.5·9-s + 55.0·11-s − 8.43·12-s + 59.6·13-s + 71.4·14-s + 16·16-s + 32.6·17-s − 45.1·18-s + 86.8·19-s − 75.3·21-s + 110.·22-s − 23·23-s − 16.8·24-s + 119.·26-s + 104.·27-s + 142.·28-s − 183.·29-s − 310.·31-s + 32·32-s − 116.·33-s + 65.2·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.405·3-s + 0.5·4-s − 0.286·6-s + 1.93·7-s + 0.353·8-s − 0.835·9-s + 1.50·11-s − 0.202·12-s + 1.27·13-s + 1.36·14-s + 0.250·16-s + 0.465·17-s − 0.590·18-s + 1.04·19-s − 0.783·21-s + 1.06·22-s − 0.208·23-s − 0.143·24-s + 0.899·26-s + 0.744·27-s + 0.965·28-s − 1.17·29-s − 1.79·31-s + 0.176·32-s − 0.612·33-s + 0.328·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1150 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(4.351121049\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.351121049\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 2T \) |
| 5 | \( 1 \) |
| 23 | \( 1 + 23T \) |
| good | 3 | \( 1 + 2.10T + 27T^{2} \) |
| 7 | \( 1 - 35.7T + 343T^{2} \) |
| 11 | \( 1 - 55.0T + 1.33e3T^{2} \) |
| 13 | \( 1 - 59.6T + 2.19e3T^{2} \) |
| 17 | \( 1 - 32.6T + 4.91e3T^{2} \) |
| 19 | \( 1 - 86.8T + 6.85e3T^{2} \) |
| 29 | \( 1 + 183.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 310.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 172.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 116.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 23.6T + 7.95e4T^{2} \) |
| 47 | \( 1 - 91.5T + 1.03e5T^{2} \) |
| 53 | \( 1 - 678.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 295.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 336.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 915.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 753.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 52.3T + 3.89e5T^{2} \) |
| 79 | \( 1 - 483.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 364.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 451.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 145.T + 9.12e5T^{2} \) |
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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
−9.233401170482975507229178824478, −8.576733319126158921574468744799, −7.71020222643901998882445147056, −6.82741233220123570357546146605, −5.59403191048622521901493399496, −5.44718172359772646186404159356, −4.17364280010763495100829318302, −3.48720271075551346329507404873, −1.85833951134499744946752771635, −1.12383034833193555121373711480,
1.12383034833193555121373711480, 1.85833951134499744946752771635, 3.48720271075551346329507404873, 4.17364280010763495100829318302, 5.44718172359772646186404159356, 5.59403191048622521901493399496, 6.82741233220123570357546146605, 7.71020222643901998882445147056, 8.576733319126158921574468744799, 9.233401170482975507229178824478
