| L(s) = 1 | + 5-s − 3·9-s + 6·11-s + 13-s + 4·19-s − 5·23-s + 25-s + 10·29-s + 5·31-s − 7·37-s − 7·41-s + 10·43-s − 3·45-s + 5·47-s + 4·53-s + 6·55-s − 6·59-s + 15·61-s + 65-s + 2·67-s − 4·71-s + 14·73-s + 10·79-s + 9·81-s + 17·83-s − 12·89-s + 4·95-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 9-s + 1.80·11-s + 0.277·13-s + 0.917·19-s − 1.04·23-s + 1/5·25-s + 1.85·29-s + 0.898·31-s − 1.15·37-s − 1.09·41-s + 1.52·43-s − 0.447·45-s + 0.729·47-s + 0.549·53-s + 0.809·55-s − 0.781·59-s + 1.92·61-s + 0.124·65-s + 0.244·67-s − 0.474·71-s + 1.63·73-s + 1.12·79-s + 81-s + 1.86·83-s − 1.27·89-s + 0.410·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 283220 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 283220 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.181725115\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.181725115\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 \) |
| 17 | \( 1 \) |
| good | 3 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 5 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 + 7 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 - 5 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 15 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - 17 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−12.46521877589347, −12.09596789606381, −12.00563115599773, −11.47387175334149, −10.88781839259313, −10.49494222209917, −9.843959163671499, −9.549101341584591, −9.054287344228273, −8.453252497219163, −8.380842245732528, −7.635012269577839, −6.942648936695877, −6.471839863946601, −6.313494903541507, −5.593669361939852, −5.244654777470250, −4.579326878663747, −3.936606492681423, −3.597173137878943, −2.928094124408498, −2.396263735134752, −1.751583588125240, −1.032986685998915, −0.6380114437977905,
0.6380114437977905, 1.032986685998915, 1.751583588125240, 2.396263735134752, 2.928094124408498, 3.597173137878943, 3.936606492681423, 4.579326878663747, 5.244654777470250, 5.593669361939852, 6.313494903541507, 6.471839863946601, 6.942648936695877, 7.635012269577839, 8.380842245732528, 8.453252497219163, 9.054287344228273, 9.549101341584591, 9.843959163671499, 10.49494222209917, 10.88781839259313, 11.47387175334149, 12.00563115599773, 12.09596789606381, 12.46521877589347
