| L(s) = 1 | + 3-s + 3.04·5-s + 9-s − 3.93·11-s + 4.88·13-s + 3.04·15-s − 5.34·17-s − 2.30·19-s + 7.93·23-s + 4.25·25-s + 27-s + 5.55·29-s − 0.645·31-s − 3.93·33-s + 5.65·37-s + 4.88·39-s + 10.0·41-s − 8.91·43-s + 3.04·45-s − 6.61·47-s − 5.34·51-s + 1.25·53-s − 11.9·55-s − 2.30·57-s + 3.04·59-s − 2.97·61-s + 14.8·65-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.36·5-s + 0.333·9-s − 1.18·11-s + 1.35·13-s + 0.785·15-s − 1.29·17-s − 0.528·19-s + 1.65·23-s + 0.851·25-s + 0.192·27-s + 1.03·29-s − 0.115·31-s − 0.684·33-s + 0.929·37-s + 0.782·39-s + 1.57·41-s − 1.35·43-s + 0.453·45-s − 0.964·47-s − 0.748·51-s + 0.172·53-s − 1.61·55-s − 0.304·57-s + 0.396·59-s − 0.380·61-s + 1.84·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4704 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4704 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.249213725\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.249213725\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 3.04T + 5T^{2} \) |
| 11 | \( 1 + 3.93T + 11T^{2} \) |
| 13 | \( 1 - 4.88T + 13T^{2} \) |
| 17 | \( 1 + 5.34T + 17T^{2} \) |
| 19 | \( 1 + 2.30T + 19T^{2} \) |
| 23 | \( 1 - 7.93T + 23T^{2} \) |
| 29 | \( 1 - 5.55T + 29T^{2} \) |
| 31 | \( 1 + 0.645T + 31T^{2} \) |
| 37 | \( 1 - 5.65T + 37T^{2} \) |
| 41 | \( 1 - 10.0T + 41T^{2} \) |
| 43 | \( 1 + 8.91T + 43T^{2} \) |
| 47 | \( 1 + 6.61T + 47T^{2} \) |
| 53 | \( 1 - 1.25T + 53T^{2} \) |
| 59 | \( 1 - 3.04T + 59T^{2} \) |
| 61 | \( 1 + 2.97T + 61T^{2} \) |
| 67 | \( 1 - 13.5T + 67T^{2} \) |
| 71 | \( 1 - 13.5T + 71T^{2} \) |
| 73 | \( 1 - 4.67T + 73T^{2} \) |
| 79 | \( 1 - 1.05T + 79T^{2} \) |
| 83 | \( 1 - 8.60T + 83T^{2} \) |
| 89 | \( 1 - 4.85T + 89T^{2} \) |
| 97 | \( 1 - 18.3T + 97T^{2} \) |
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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
−8.442906910010638729512196016447, −7.71382769679460992180328481696, −6.50584039241657973672594562388, −6.41632736890333436956397703411, −5.28775670201575671193204214118, −4.74244104947422489230276713314, −3.62996158578987141425427802680, −2.64839904851103242357005260168, −2.13920170265743500435690199180, −1.00632302306326863466282982463,
1.00632302306326863466282982463, 2.13920170265743500435690199180, 2.64839904851103242357005260168, 3.62996158578987141425427802680, 4.74244104947422489230276713314, 5.28775670201575671193204214118, 6.41632736890333436956397703411, 6.50584039241657973672594562388, 7.71382769679460992180328481696, 8.442906910010638729512196016447
