| L(s) = 1 | − 2-s + 4-s − 3·5-s + 4·7-s − 8-s + 3·10-s + 3·11-s + 2·13-s − 4·14-s + 16-s + 8·19-s − 3·20-s − 3·22-s + 6·23-s + 4·25-s − 2·26-s + 4·28-s + 3·29-s + 7·31-s − 32-s − 12·35-s − 8·37-s − 8·38-s + 3·40-s + 6·41-s − 4·43-s + 3·44-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 1.34·5-s + 1.51·7-s − 0.353·8-s + 0.948·10-s + 0.904·11-s + 0.554·13-s − 1.06·14-s + 1/4·16-s + 1.83·19-s − 0.670·20-s − 0.639·22-s + 1.25·23-s + 4/5·25-s − 0.392·26-s + 0.755·28-s + 0.557·29-s + 1.25·31-s − 0.176·32-s − 2.02·35-s − 1.31·37-s − 1.29·38-s + 0.474·40-s + 0.937·41-s − 0.609·43-s + 0.452·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5202 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5202 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.658447503\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.658447503\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 17 | \( 1 \) |
| good | 5 | \( 1 + 3 T + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 - 7 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 + 15 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 7 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 13 T + p T^{2} \) |
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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
−8.235178303377333794500016101573, −7.50963224871798970165971124923, −7.21386475613061496824087437437, −6.20573888458528416164034907418, −5.15578659679404072792425092812, −4.53682872021803001666891978218, −3.67106318635667668728499397153, −2.88584426553706329935745966433, −1.45284542183315962793965839781, −0.899228488521222409797104317853,
0.899228488521222409797104317853, 1.45284542183315962793965839781, 2.88584426553706329935745966433, 3.67106318635667668728499397153, 4.53682872021803001666891978218, 5.15578659679404072792425092812, 6.20573888458528416164034907418, 7.21386475613061496824087437437, 7.50963224871798970165971124923, 8.235178303377333794500016101573
