| L(s) = 1 | − 3-s + 2.76·7-s + 9-s + 4.49·11-s + 13-s + 3.62·17-s − 6.14·19-s − 2.76·21-s + 5.52·23-s − 27-s + 6.25·29-s + 8.49·31-s − 4.49·33-s − 0.896·37-s − 39-s + 2.89·41-s + 0.270·43-s + 3.38·47-s + 0.626·49-s − 3.62·51-s − 3.27·53-s + 6.14·57-s − 9.53·59-s + 11.4·61-s + 2.76·63-s − 12.4·67-s − 5.52·69-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.04·7-s + 0.333·9-s + 1.35·11-s + 0.277·13-s + 0.879·17-s − 1.41·19-s − 0.602·21-s + 1.15·23-s − 0.192·27-s + 1.16·29-s + 1.52·31-s − 0.781·33-s − 0.147·37-s − 0.160·39-s + 0.452·41-s + 0.0412·43-s + 0.494·47-s + 0.0894·49-s − 0.507·51-s − 0.449·53-s + 0.814·57-s − 1.24·59-s + 1.46·61-s + 0.347·63-s − 1.51·67-s − 0.664·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.432650886\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.432650886\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 - 2.76T + 7T^{2} \) |
| 11 | \( 1 - 4.49T + 11T^{2} \) |
| 17 | \( 1 - 3.62T + 17T^{2} \) |
| 19 | \( 1 + 6.14T + 19T^{2} \) |
| 23 | \( 1 - 5.52T + 23T^{2} \) |
| 29 | \( 1 - 6.25T + 29T^{2} \) |
| 31 | \( 1 - 8.49T + 31T^{2} \) |
| 37 | \( 1 + 0.896T + 37T^{2} \) |
| 41 | \( 1 - 2.89T + 41T^{2} \) |
| 43 | \( 1 - 0.270T + 43T^{2} \) |
| 47 | \( 1 - 3.38T + 47T^{2} \) |
| 53 | \( 1 + 3.27T + 53T^{2} \) |
| 59 | \( 1 + 9.53T + 59T^{2} \) |
| 61 | \( 1 - 11.4T + 61T^{2} \) |
| 67 | \( 1 + 12.4T + 67T^{2} \) |
| 71 | \( 1 - 8.89T + 71T^{2} \) |
| 73 | \( 1 - 3.25T + 73T^{2} \) |
| 79 | \( 1 - 13.6T + 79T^{2} \) |
| 83 | \( 1 + 3.03T + 83T^{2} \) |
| 89 | \( 1 + 2.20T + 89T^{2} \) |
| 97 | \( 1 + 17.7T + 97T^{2} \) |
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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
−7.983558325108707744759040585001, −6.97688020997106305419441151109, −6.48656465712215944311845932937, −5.85785052532985411325202301245, −4.89801923147723076273300320232, −4.48139705919317820498532377454, −3.69453926412329594451076877470, −2.62199306089391723577718638440, −1.50108316716807401137140963875, −0.907818607186173803066150140910,
0.907818607186173803066150140910, 1.50108316716807401137140963875, 2.62199306089391723577718638440, 3.69453926412329594451076877470, 4.48139705919317820498532377454, 4.89801923147723076273300320232, 5.85785052532985411325202301245, 6.48656465712215944311845932937, 6.97688020997106305419441151109, 7.983558325108707744759040585001
