| L(s) = 1 | + 1.69·3-s − 4.22·7-s − 0.118·9-s − 4.59·11-s − 0.978·13-s − 3.04·17-s − 1.91·19-s − 7.17·21-s − 23-s − 5.29·27-s − 0.737·29-s − 2.97·31-s − 7.79·33-s + 8.93·37-s − 1.66·39-s + 9.08·41-s + 6.97·43-s − 2.58·47-s + 10.8·49-s − 5.16·51-s − 2.71·53-s − 3.25·57-s − 7.13·59-s − 0.731·61-s + 0.500·63-s + 7.16·67-s − 1.69·69-s + ⋯ |
| L(s) = 1 | + 0.980·3-s − 1.59·7-s − 0.0394·9-s − 1.38·11-s − 0.271·13-s − 0.737·17-s − 0.439·19-s − 1.56·21-s − 0.208·23-s − 1.01·27-s − 0.136·29-s − 0.535·31-s − 1.35·33-s + 1.46·37-s − 0.265·39-s + 1.41·41-s + 1.06·43-s − 0.377·47-s + 1.55·49-s − 0.722·51-s − 0.372·53-s − 0.430·57-s − 0.928·59-s − 0.0936·61-s + 0.0630·63-s + 0.875·67-s − 0.204·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.118321127\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.118321127\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 3 | \( 1 - 1.69T + 3T^{2} \) |
| 7 | \( 1 + 4.22T + 7T^{2} \) |
| 11 | \( 1 + 4.59T + 11T^{2} \) |
| 13 | \( 1 + 0.978T + 13T^{2} \) |
| 17 | \( 1 + 3.04T + 17T^{2} \) |
| 19 | \( 1 + 1.91T + 19T^{2} \) |
| 29 | \( 1 + 0.737T + 29T^{2} \) |
| 31 | \( 1 + 2.97T + 31T^{2} \) |
| 37 | \( 1 - 8.93T + 37T^{2} \) |
| 41 | \( 1 - 9.08T + 41T^{2} \) |
| 43 | \( 1 - 6.97T + 43T^{2} \) |
| 47 | \( 1 + 2.58T + 47T^{2} \) |
| 53 | \( 1 + 2.71T + 53T^{2} \) |
| 59 | \( 1 + 7.13T + 59T^{2} \) |
| 61 | \( 1 + 0.731T + 61T^{2} \) |
| 67 | \( 1 - 7.16T + 67T^{2} \) |
| 71 | \( 1 - 6.08T + 71T^{2} \) |
| 73 | \( 1 + 5.96T + 73T^{2} \) |
| 79 | \( 1 - 2.06T + 79T^{2} \) |
| 83 | \( 1 + 4.39T + 83T^{2} \) |
| 89 | \( 1 + 7.25T + 89T^{2} \) |
| 97 | \( 1 + 7.04T + 97T^{2} \) |
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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
−7.58409795133232755055621688070, −7.34405734428360306873058481314, −6.15513440761254614968287882082, −5.97342339774083888561826673837, −4.86490844277693394301361727399, −4.03006922797881426209433519148, −3.26566399493246748346635531649, −2.63180379982313750705778750864, −2.20469542513516051878929538679, −0.44171178511265067104786139200,
0.44171178511265067104786139200, 2.20469542513516051878929538679, 2.63180379982313750705778750864, 3.26566399493246748346635531649, 4.03006922797881426209433519148, 4.86490844277693394301361727399, 5.97342339774083888561826673837, 6.15513440761254614968287882082, 7.34405734428360306873058481314, 7.58409795133232755055621688070
