| L(s) = 1 | − 2.79·3-s + 1.79·5-s + 4.79·9-s − 3.58·11-s + 4·13-s − 5·15-s + 17-s + 2·19-s + 1.58·23-s − 1.79·25-s − 4.99·27-s + 0.208·31-s + 10·33-s + 5.58·37-s − 11.1·39-s + 1.20·41-s − 2.79·43-s + 8.58·45-s − 1.58·47-s − 2.79·51-s − 9.79·53-s − 6.41·55-s − 5.58·57-s + 11.5·59-s + 6.79·61-s + 7.16·65-s − 10.9·67-s + ⋯ |
| L(s) = 1 | − 1.61·3-s + 0.801·5-s + 1.59·9-s − 1.08·11-s + 1.10·13-s − 1.29·15-s + 0.242·17-s + 0.458·19-s + 0.329·23-s − 0.358·25-s − 0.962·27-s + 0.0374·31-s + 1.74·33-s + 0.917·37-s − 1.78·39-s + 0.188·41-s − 0.425·43-s + 1.27·45-s − 0.230·47-s − 0.390·51-s − 1.34·53-s − 0.865·55-s − 0.739·57-s + 1.50·59-s + 0.869·61-s + 0.888·65-s − 1.33·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.131309952\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.131309952\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 3 | \( 1 + 2.79T + 3T^{2} \) |
| 5 | \( 1 - 1.79T + 5T^{2} \) |
| 11 | \( 1 + 3.58T + 11T^{2} \) |
| 13 | \( 1 - 4T + 13T^{2} \) |
| 19 | \( 1 - 2T + 19T^{2} \) |
| 23 | \( 1 - 1.58T + 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 - 0.208T + 31T^{2} \) |
| 37 | \( 1 - 5.58T + 37T^{2} \) |
| 41 | \( 1 - 1.20T + 41T^{2} \) |
| 43 | \( 1 + 2.79T + 43T^{2} \) |
| 47 | \( 1 + 1.58T + 47T^{2} \) |
| 53 | \( 1 + 9.79T + 53T^{2} \) |
| 59 | \( 1 - 11.5T + 59T^{2} \) |
| 61 | \( 1 - 6.79T + 61T^{2} \) |
| 67 | \( 1 + 10.9T + 67T^{2} \) |
| 71 | \( 1 - 4.41T + 71T^{2} \) |
| 73 | \( 1 - 14.7T + 73T^{2} \) |
| 79 | \( 1 + 10T + 79T^{2} \) |
| 83 | \( 1 + 9.58T + 83T^{2} \) |
| 89 | \( 1 + 11.1T + 89T^{2} \) |
| 97 | \( 1 - 9.79T + 97T^{2} \) |
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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
−8.598539873270534062537199283794, −7.77834820253630330623211250396, −6.88163862885747332863183352606, −6.16561685839038769275965072916, −5.62438240955602658525752807987, −5.13728492055850350978206559993, −4.20567628109388525966213488019, −3.00985045279199418994643201030, −1.75567212017422970424995226382, −0.70641798711397202568177346921,
0.70641798711397202568177346921, 1.75567212017422970424995226382, 3.00985045279199418994643201030, 4.20567628109388525966213488019, 5.13728492055850350978206559993, 5.62438240955602658525752807987, 6.16561685839038769275965072916, 6.88163862885747332863183352606, 7.77834820253630330623211250396, 8.598539873270534062537199283794
