| L(s) = 1 | − 3-s + 2.74·5-s + 9-s + 1.54·11-s + 6.03·13-s − 2.74·15-s − 7.49·17-s + 6.03·19-s + 7.49·23-s + 2.54·25-s − 27-s − 1.25·29-s + 5.29·31-s − 1.54·33-s − 4.94·37-s − 6.03·39-s + 5.08·41-s + 3.45·43-s + 2.74·45-s + 9.49·47-s + 7.49·51-s + 3.83·53-s + 4.23·55-s − 6.03·57-s − 5.54·59-s − 14.5·61-s + 16.5·65-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.22·5-s + 0.333·9-s + 0.465·11-s + 1.67·13-s − 0.709·15-s − 1.81·17-s + 1.38·19-s + 1.56·23-s + 0.508·25-s − 0.192·27-s − 0.232·29-s + 0.950·31-s − 0.268·33-s − 0.813·37-s − 0.966·39-s + 0.794·41-s + 0.527·43-s + 0.409·45-s + 1.38·47-s + 1.04·51-s + 0.526·53-s + 0.571·55-s − 0.799·57-s − 0.721·59-s − 1.86·61-s + 2.05·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9408 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9408 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.740383111\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.740383111\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 2.74T + 5T^{2} \) |
| 11 | \( 1 - 1.54T + 11T^{2} \) |
| 13 | \( 1 - 6.03T + 13T^{2} \) |
| 17 | \( 1 + 7.49T + 17T^{2} \) |
| 19 | \( 1 - 6.03T + 19T^{2} \) |
| 23 | \( 1 - 7.49T + 23T^{2} \) |
| 29 | \( 1 + 1.25T + 29T^{2} \) |
| 31 | \( 1 - 5.29T + 31T^{2} \) |
| 37 | \( 1 + 4.94T + 37T^{2} \) |
| 41 | \( 1 - 5.08T + 41T^{2} \) |
| 43 | \( 1 - 3.45T + 43T^{2} \) |
| 47 | \( 1 - 9.49T + 47T^{2} \) |
| 53 | \( 1 - 3.83T + 53T^{2} \) |
| 59 | \( 1 + 5.54T + 59T^{2} \) |
| 61 | \( 1 + 14.5T + 61T^{2} \) |
| 67 | \( 1 + 4.03T + 67T^{2} \) |
| 71 | \( 1 - 5.49T + 71T^{2} \) |
| 73 | \( 1 + 12.5T + 73T^{2} \) |
| 79 | \( 1 - 7.79T + 79T^{2} \) |
| 83 | \( 1 - 6.52T + 83T^{2} \) |
| 89 | \( 1 - 9.49T + 89T^{2} \) |
| 97 | \( 1 + 1.54T + 97T^{2} \) |
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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
−7.50538020083450507117646525401, −6.83703970916781063670435219124, −6.19012410509135295958823454038, −5.85797746772003044989683628698, −5.02811541677113932625704160923, −4.34998824963240448053312510011, −3.43122749814822771684727054663, −2.52847211601382724989689382595, −1.55022770192469998420233780204, −0.900993837304344759186486518542,
0.900993837304344759186486518542, 1.55022770192469998420233780204, 2.52847211601382724989689382595, 3.43122749814822771684727054663, 4.34998824963240448053312510011, 5.02811541677113932625704160923, 5.85797746772003044989683628698, 6.19012410509135295958823454038, 6.83703970916781063670435219124, 7.50538020083450507117646525401
