| L(s) = 1 | + 5-s + 3.44·7-s + 2·17-s − 6.89·19-s + 7.44·23-s + 25-s + 1.89·29-s + 1.10·31-s + 3.44·35-s − 6·37-s + 9.89·41-s + 11.7·43-s + 9.44·47-s + 4.89·49-s − 7.79·53-s + 1.10·59-s − 3·61-s − 13.2·67-s − 9.79·71-s + 13.7·73-s − 6.89·79-s + 5.44·83-s + 2·85-s − 2.79·89-s − 6.89·95-s + 2·97-s + 2·101-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 1.30·7-s + 0.485·17-s − 1.58·19-s + 1.55·23-s + 0.200·25-s + 0.352·29-s + 0.197·31-s + 0.583·35-s − 0.986·37-s + 1.54·41-s + 1.79·43-s + 1.37·47-s + 0.699·49-s − 1.07·53-s + 0.143·59-s − 0.384·61-s − 1.61·67-s − 1.16·71-s + 1.61·73-s − 0.776·79-s + 0.598·83-s + 0.216·85-s − 0.296·89-s − 0.707·95-s + 0.203·97-s + 0.199·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.497801484\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.497801484\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| good | 7 | \( 1 - 3.44T + 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 - 2T + 17T^{2} \) |
| 19 | \( 1 + 6.89T + 19T^{2} \) |
| 23 | \( 1 - 7.44T + 23T^{2} \) |
| 29 | \( 1 - 1.89T + 29T^{2} \) |
| 31 | \( 1 - 1.10T + 31T^{2} \) |
| 37 | \( 1 + 6T + 37T^{2} \) |
| 41 | \( 1 - 9.89T + 41T^{2} \) |
| 43 | \( 1 - 11.7T + 43T^{2} \) |
| 47 | \( 1 - 9.44T + 47T^{2} \) |
| 53 | \( 1 + 7.79T + 53T^{2} \) |
| 59 | \( 1 - 1.10T + 59T^{2} \) |
| 61 | \( 1 + 3T + 61T^{2} \) |
| 67 | \( 1 + 13.2T + 67T^{2} \) |
| 71 | \( 1 + 9.79T + 71T^{2} \) |
| 73 | \( 1 - 13.7T + 73T^{2} \) |
| 79 | \( 1 + 6.89T + 79T^{2} \) |
| 83 | \( 1 - 5.44T + 83T^{2} \) |
| 89 | \( 1 + 2.79T + 89T^{2} \) |
| 97 | \( 1 - 2T + 97T^{2} \) |
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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
−8.810094473244902225863958671273, −7.86872472112905429783454606288, −7.29561105727588080836615868616, −6.32629228830983142066046800463, −5.59886534851593283043038125112, −4.76715643472403162551356374357, −4.18775743972261135793208126128, −2.88796835386999305826815644374, −2.00506994364080859634970460265, −1.01004956177176863900695628427,
1.01004956177176863900695628427, 2.00506994364080859634970460265, 2.88796835386999305826815644374, 4.18775743972261135793208126128, 4.76715643472403162551356374357, 5.59886534851593283043038125112, 6.32629228830983142066046800463, 7.29561105727588080836615868616, 7.86872472112905429783454606288, 8.810094473244902225863958671273
