| L(s) = 1 | + 3-s + 0.0593·5-s + 1.53·7-s + 9-s − 0.726·11-s − 1.12·13-s + 0.0593·15-s − 6.50·17-s + 4.67·19-s + 1.53·21-s + 3.23·23-s − 4.99·25-s + 27-s − 3.01·29-s + 0.738·31-s − 0.726·33-s + 0.0911·35-s + 8.63·37-s − 1.12·39-s + 10.3·41-s − 7.89·43-s + 0.0593·45-s + 1.39·47-s − 4.64·49-s − 6.50·51-s + 13.9·53-s − 0.0431·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.0265·5-s + 0.580·7-s + 0.333·9-s − 0.219·11-s − 0.311·13-s + 0.0153·15-s − 1.57·17-s + 1.07·19-s + 0.334·21-s + 0.674·23-s − 0.999·25-s + 0.192·27-s − 0.559·29-s + 0.132·31-s − 0.126·33-s + 0.0153·35-s + 1.42·37-s − 0.179·39-s + 1.62·41-s − 1.20·43-s + 0.00884·45-s + 0.203·47-s − 0.663·49-s − 0.911·51-s + 1.91·53-s − 0.00581·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.634318632\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.634318632\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 167 | \( 1 - T \) |
| good | 5 | \( 1 - 0.0593T + 5T^{2} \) |
| 7 | \( 1 - 1.53T + 7T^{2} \) |
| 11 | \( 1 + 0.726T + 11T^{2} \) |
| 13 | \( 1 + 1.12T + 13T^{2} \) |
| 17 | \( 1 + 6.50T + 17T^{2} \) |
| 19 | \( 1 - 4.67T + 19T^{2} \) |
| 23 | \( 1 - 3.23T + 23T^{2} \) |
| 29 | \( 1 + 3.01T + 29T^{2} \) |
| 31 | \( 1 - 0.738T + 31T^{2} \) |
| 37 | \( 1 - 8.63T + 37T^{2} \) |
| 41 | \( 1 - 10.3T + 41T^{2} \) |
| 43 | \( 1 + 7.89T + 43T^{2} \) |
| 47 | \( 1 - 1.39T + 47T^{2} \) |
| 53 | \( 1 - 13.9T + 53T^{2} \) |
| 59 | \( 1 - 6.92T + 59T^{2} \) |
| 61 | \( 1 + 13.9T + 61T^{2} \) |
| 67 | \( 1 - 8.78T + 67T^{2} \) |
| 71 | \( 1 - 9.28T + 71T^{2} \) |
| 73 | \( 1 - 9.04T + 73T^{2} \) |
| 79 | \( 1 - 9.62T + 79T^{2} \) |
| 83 | \( 1 + 6.72T + 83T^{2} \) |
| 89 | \( 1 - 7.03T + 89T^{2} \) |
| 97 | \( 1 + 0.754T + 97T^{2} \) |
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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
−7.83756521116977316880889894989, −7.29305614372930793329499615660, −6.56456408609904178977456227658, −5.70028945317833095499937068912, −4.93108186633477316991681511898, −4.30822886135058043771822951719, −3.51010394046450360684853664127, −2.53080391314980104766952597997, −1.97983712832801252495747625962, −0.77760175403011349166285644170,
0.77760175403011349166285644170, 1.97983712832801252495747625962, 2.53080391314980104766952597997, 3.51010394046450360684853664127, 4.30822886135058043771822951719, 4.93108186633477316991681511898, 5.70028945317833095499937068912, 6.56456408609904178977456227658, 7.29305614372930793329499615660, 7.83756521116977316880889894989
