| L(s) = 1 | − 1.83·3-s + 1.83·7-s + 0.364·9-s − 0.834·11-s − 2.19·13-s + 2.56·17-s + 19-s − 3.36·21-s − 0.635·23-s + 4.83·27-s − 9.62·29-s + 6.59·31-s + 1.53·33-s − 5.23·37-s + 4.03·39-s + 4.43·41-s − 7.06·43-s + 9.86·47-s − 3.63·49-s − 4.70·51-s − 0.668·53-s − 1.83·57-s − 0.397·59-s + 2.26·61-s + 0.668·63-s + 2.43·67-s + 1.16·69-s + ⋯ |
| L(s) = 1 | − 1.05·3-s + 0.693·7-s + 0.121·9-s − 0.251·11-s − 0.609·13-s + 0.621·17-s + 0.229·19-s − 0.734·21-s − 0.132·23-s + 0.930·27-s − 1.78·29-s + 1.18·31-s + 0.266·33-s − 0.860·37-s + 0.645·39-s + 0.691·41-s − 1.07·43-s + 1.43·47-s − 0.519·49-s − 0.658·51-s − 0.0918·53-s − 0.242·57-s − 0.0517·59-s + 0.289·61-s + 0.0842·63-s + 0.297·67-s + 0.140·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.085843967\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.085843967\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 + 1.83T + 3T^{2} \) |
| 7 | \( 1 - 1.83T + 7T^{2} \) |
| 11 | \( 1 + 0.834T + 11T^{2} \) |
| 13 | \( 1 + 2.19T + 13T^{2} \) |
| 17 | \( 1 - 2.56T + 17T^{2} \) |
| 23 | \( 1 + 0.635T + 23T^{2} \) |
| 29 | \( 1 + 9.62T + 29T^{2} \) |
| 31 | \( 1 - 6.59T + 31T^{2} \) |
| 37 | \( 1 + 5.23T + 37T^{2} \) |
| 41 | \( 1 - 4.43T + 41T^{2} \) |
| 43 | \( 1 + 7.06T + 43T^{2} \) |
| 47 | \( 1 - 9.86T + 47T^{2} \) |
| 53 | \( 1 + 0.668T + 53T^{2} \) |
| 59 | \( 1 + 0.397T + 59T^{2} \) |
| 61 | \( 1 - 2.26T + 61T^{2} \) |
| 67 | \( 1 - 2.43T + 67T^{2} \) |
| 71 | \( 1 + 4.12T + 71T^{2} \) |
| 73 | \( 1 - 9.49T + 73T^{2} \) |
| 79 | \( 1 + 2.62T + 79T^{2} \) |
| 83 | \( 1 - 10.8T + 83T^{2} \) |
| 89 | \( 1 + 5.97T + 89T^{2} \) |
| 97 | \( 1 - 10.5T + 97T^{2} \) |
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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
−7.80092233355672908222194904705, −7.17508040309830367165537760936, −6.40228643086464313250830305434, −5.61453006491792719253143023063, −5.22247068409608625178184599529, −4.56211357481779345236908163252, −3.62180664630984072952840358602, −2.62801746698960412056659645496, −1.64343460817352830999490658446, −0.55778423598784751467143217399,
0.55778423598784751467143217399, 1.64343460817352830999490658446, 2.62801746698960412056659645496, 3.62180664630984072952840358602, 4.56211357481779345236908163252, 5.22247068409608625178184599529, 5.61453006491792719253143023063, 6.40228643086464313250830305434, 7.17508040309830367165537760936, 7.80092233355672908222194904705
