| L(s) = 1 | + 3-s + 0.920·5-s + 4.87·7-s + 9-s + 0.964·11-s + 0.920·15-s + 2.47·17-s − 3.12·19-s + 4.87·21-s − 4.92·23-s − 4.15·25-s + 27-s + 6.90·29-s + 4.61·31-s + 0.964·33-s + 4.48·35-s − 4.81·37-s + 3.43·41-s + 10.0·43-s + 0.920·45-s + 8.64·47-s + 16.7·49-s + 2.47·51-s + 0.841·53-s + 0.887·55-s − 3.12·57-s + 9.35·59-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.411·5-s + 1.84·7-s + 0.333·9-s + 0.290·11-s + 0.237·15-s + 0.600·17-s − 0.717·19-s + 1.06·21-s − 1.02·23-s − 0.830·25-s + 0.192·27-s + 1.28·29-s + 0.828·31-s + 0.167·33-s + 0.757·35-s − 0.791·37-s + 0.536·41-s + 1.53·43-s + 0.137·45-s + 1.26·47-s + 2.38·49-s + 0.346·51-s + 0.115·53-s + 0.119·55-s − 0.414·57-s + 1.21·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4056 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4056 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.468680493\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.468680493\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 - 0.920T + 5T^{2} \) |
| 7 | \( 1 - 4.87T + 7T^{2} \) |
| 11 | \( 1 - 0.964T + 11T^{2} \) |
| 17 | \( 1 - 2.47T + 17T^{2} \) |
| 19 | \( 1 + 3.12T + 19T^{2} \) |
| 23 | \( 1 + 4.92T + 23T^{2} \) |
| 29 | \( 1 - 6.90T + 29T^{2} \) |
| 31 | \( 1 - 4.61T + 31T^{2} \) |
| 37 | \( 1 + 4.81T + 37T^{2} \) |
| 41 | \( 1 - 3.43T + 41T^{2} \) |
| 43 | \( 1 - 10.0T + 43T^{2} \) |
| 47 | \( 1 - 8.64T + 47T^{2} \) |
| 53 | \( 1 - 0.841T + 53T^{2} \) |
| 59 | \( 1 - 9.35T + 59T^{2} \) |
| 61 | \( 1 - 0.793T + 61T^{2} \) |
| 67 | \( 1 + 6.99T + 67T^{2} \) |
| 71 | \( 1 + 15.5T + 71T^{2} \) |
| 73 | \( 1 + 5.72T + 73T^{2} \) |
| 79 | \( 1 + 13.8T + 79T^{2} \) |
| 83 | \( 1 + 14.8T + 83T^{2} \) |
| 89 | \( 1 - 10.6T + 89T^{2} \) |
| 97 | \( 1 + 9.53T + 97T^{2} \) |
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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
−8.526289127876816735679938067603, −7.76773591392835734737130329205, −7.24668799902829038882465179232, −6.08715595969939084023914881421, −5.50763751615366492335970997140, −4.43301292134159690283972825700, −4.12475857197434089298613028303, −2.72488159055658648424516065793, −1.96466226436294689279101798076, −1.14935220276289745643211724090,
1.14935220276289745643211724090, 1.96466226436294689279101798076, 2.72488159055658648424516065793, 4.12475857197434089298613028303, 4.43301292134159690283972825700, 5.50763751615366492335970997140, 6.08715595969939084023914881421, 7.24668799902829038882465179232, 7.76773591392835734737130329205, 8.526289127876816735679938067603
