| L(s) = 1 | − 1.87·2-s + 1.52·4-s − 0.769·5-s − 0.850·7-s + 0.900·8-s + 1.44·10-s − 11-s − 0.486·13-s + 1.59·14-s − 4.72·16-s − 2.67·17-s − 7.96·19-s − 1.16·20-s + 1.87·22-s − 7.17·23-s − 4.40·25-s + 0.912·26-s − 1.29·28-s + 4.07·29-s − 7.87·31-s + 7.07·32-s + 5.02·34-s + 0.653·35-s + 2.50·37-s + 14.9·38-s − 0.692·40-s − 1.48·41-s + ⋯ |
| L(s) = 1 | − 1.32·2-s + 0.760·4-s − 0.343·5-s − 0.321·7-s + 0.318·8-s + 0.456·10-s − 0.301·11-s − 0.134·13-s + 0.426·14-s − 1.18·16-s − 0.649·17-s − 1.82·19-s − 0.261·20-s + 0.400·22-s − 1.49·23-s − 0.881·25-s + 0.179·26-s − 0.244·28-s + 0.755·29-s − 1.41·31-s + 1.25·32-s + 0.861·34-s + 0.110·35-s + 0.411·37-s + 2.42·38-s − 0.109·40-s − 0.232·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1917128502\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1917128502\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 + T \) |
| 61 | \( 1 - T \) |
| good | 2 | \( 1 + 1.87T + 2T^{2} \) |
| 5 | \( 1 + 0.769T + 5T^{2} \) |
| 7 | \( 1 + 0.850T + 7T^{2} \) |
| 13 | \( 1 + 0.486T + 13T^{2} \) |
| 17 | \( 1 + 2.67T + 17T^{2} \) |
| 19 | \( 1 + 7.96T + 19T^{2} \) |
| 23 | \( 1 + 7.17T + 23T^{2} \) |
| 29 | \( 1 - 4.07T + 29T^{2} \) |
| 31 | \( 1 + 7.87T + 31T^{2} \) |
| 37 | \( 1 - 2.50T + 37T^{2} \) |
| 41 | \( 1 + 1.48T + 41T^{2} \) |
| 43 | \( 1 + 6.73T + 43T^{2} \) |
| 47 | \( 1 + 12.0T + 47T^{2} \) |
| 53 | \( 1 - 11.5T + 53T^{2} \) |
| 59 | \( 1 - 4.12T + 59T^{2} \) |
| 67 | \( 1 - 6.50T + 67T^{2} \) |
| 71 | \( 1 - 7.19T + 71T^{2} \) |
| 73 | \( 1 + 7.31T + 73T^{2} \) |
| 79 | \( 1 - 8.91T + 79T^{2} \) |
| 83 | \( 1 - 9.65T + 83T^{2} \) |
| 89 | \( 1 + 7.33T + 89T^{2} \) |
| 97 | \( 1 - 5.47T + 97T^{2} \) |
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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
−8.178733759234264287681229776264, −7.68881571897194810822167486216, −6.71192924378626291821233329028, −6.36378876644966251673599485055, −5.21864467358688934022875179903, −4.31913923742533008069312333009, −3.69642836110758363222738162142, −2.33618061675217056340803961045, −1.79436529383457711688293418122, −0.26941339608931319569775312468,
0.26941339608931319569775312468, 1.79436529383457711688293418122, 2.33618061675217056340803961045, 3.69642836110758363222738162142, 4.31913923742533008069312333009, 5.21864467358688934022875179903, 6.36378876644966251673599485055, 6.71192924378626291821233329028, 7.68881571897194810822167486216, 8.178733759234264287681229776264
