| L(s) = 1 | + 2.73·5-s + 7-s + 2.24·11-s − 0.458·13-s + 7.44·17-s + 4.24·19-s + 4.70·23-s + 2.50·25-s + 1.73·29-s − 1.73·31-s + 2.73·35-s + 2.49·37-s + 4.73·41-s − 1.96·43-s − 9.90·47-s + 49-s − 5.42·53-s + 6.15·55-s − 6.47·59-s + 5.23·61-s − 1.25·65-s − 2.77·67-s − 0.703·71-s − 7.96·73-s + 2.24·77-s − 6.66·79-s − 3.50·83-s + ⋯ |
| L(s) = 1 | + 1.22·5-s + 0.377·7-s + 0.676·11-s − 0.127·13-s + 1.80·17-s + 0.973·19-s + 0.980·23-s + 0.501·25-s + 0.323·29-s − 0.312·31-s + 0.463·35-s + 0.410·37-s + 0.740·41-s − 0.299·43-s − 1.44·47-s + 0.142·49-s − 0.744·53-s + 0.829·55-s − 0.843·59-s + 0.670·61-s − 0.155·65-s − 0.339·67-s − 0.0835·71-s − 0.932·73-s + 0.255·77-s − 0.750·79-s − 0.384·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.304066208\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.304066208\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 5 | \( 1 - 2.73T + 5T^{2} \) |
| 11 | \( 1 - 2.24T + 11T^{2} \) |
| 13 | \( 1 + 0.458T + 13T^{2} \) |
| 17 | \( 1 - 7.44T + 17T^{2} \) |
| 19 | \( 1 - 4.24T + 19T^{2} \) |
| 23 | \( 1 - 4.70T + 23T^{2} \) |
| 29 | \( 1 - 1.73T + 29T^{2} \) |
| 31 | \( 1 + 1.73T + 31T^{2} \) |
| 37 | \( 1 - 2.49T + 37T^{2} \) |
| 41 | \( 1 - 4.73T + 41T^{2} \) |
| 43 | \( 1 + 1.96T + 43T^{2} \) |
| 47 | \( 1 + 9.90T + 47T^{2} \) |
| 53 | \( 1 + 5.42T + 53T^{2} \) |
| 59 | \( 1 + 6.47T + 59T^{2} \) |
| 61 | \( 1 - 5.23T + 61T^{2} \) |
| 67 | \( 1 + 2.77T + 67T^{2} \) |
| 71 | \( 1 + 0.703T + 71T^{2} \) |
| 73 | \( 1 + 7.96T + 73T^{2} \) |
| 79 | \( 1 + 6.66T + 79T^{2} \) |
| 83 | \( 1 + 3.50T + 83T^{2} \) |
| 89 | \( 1 - 1.78T + 89T^{2} \) |
| 97 | \( 1 - 13.1T + 97T^{2} \) |
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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
−7.981907949627222535139901060755, −7.40984215851139510579523782203, −6.57457778781095266118858154241, −5.84929879268968449479614102728, −5.33113205950624454297254971020, −4.62058999487592857434156804450, −3.46896820614003616419620174203, −2.82119765943172021065744862644, −1.65759641592799267567329070426, −1.08163618158566882270727403341,
1.08163618158566882270727403341, 1.65759641592799267567329070426, 2.82119765943172021065744862644, 3.46896820614003616419620174203, 4.62058999487592857434156804450, 5.33113205950624454297254971020, 5.84929879268968449479614102728, 6.57457778781095266118858154241, 7.40984215851139510579523782203, 7.981907949627222535139901060755
