| L(s) = 1 | + 3-s − 2·7-s + 9-s − 2·11-s − 13-s − 7·17-s − 2·21-s − 3·23-s − 5·25-s + 27-s + 5·31-s − 2·33-s − 3·37-s − 39-s − 3·41-s − 11·43-s − 3·49-s − 7·51-s + 6·53-s − 13·59-s + 11·61-s − 2·63-s − 7·67-s − 3·69-s − 8·71-s − 16·73-s − 5·75-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.755·7-s + 1/3·9-s − 0.603·11-s − 0.277·13-s − 1.69·17-s − 0.436·21-s − 0.625·23-s − 25-s + 0.192·27-s + 0.898·31-s − 0.348·33-s − 0.493·37-s − 0.160·39-s − 0.468·41-s − 1.67·43-s − 3/7·49-s − 0.980·51-s + 0.824·53-s − 1.69·59-s + 1.40·61-s − 0.251·63-s − 0.855·67-s − 0.361·69-s − 0.949·71-s − 1.87·73-s − 0.577·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 112632 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112632 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| 19 | \( 1 \) |
| good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 7 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 + 11 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 13 T + p T^{2} \) |
| 61 | \( 1 - 11 T + p T^{2} \) |
| 67 | \( 1 + 7 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 16 T + p T^{2} \) |
| 79 | \( 1 + 2 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + T + p T^{2} \) |
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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
−13.95595924203434, −13.47431779899160, −13.27914838035974, −12.90754975394239, −12.03551073314032, −11.88076997901212, −11.22068799254155, −10.52131284021980, −10.17618779127312, −9.750928350834329, −9.198286687232087, −8.680338003861271, −8.241474601464504, −7.755394077445383, −7.043815998221464, −6.725919222508845, −6.140499254163530, −5.588208349126356, −4.825080587389474, −4.375362128389926, −3.807372205440162, −3.111141133338846, −2.654633457287310, −2.035073965461569, −1.429224707211243, 0, 0,
1.429224707211243, 2.035073965461569, 2.654633457287310, 3.111141133338846, 3.807372205440162, 4.375362128389926, 4.825080587389474, 5.588208349126356, 6.140499254163530, 6.725919222508845, 7.043815998221464, 7.755394077445383, 8.241474601464504, 8.680338003861271, 9.198286687232087, 9.750928350834329, 10.17618779127312, 10.52131284021980, 11.22068799254155, 11.88076997901212, 12.03551073314032, 12.90754975394239, 13.27914838035974, 13.47431779899160, 13.95595924203434
