| L(s) = 1 | − 3·3-s + 5·5-s + 7·7-s + 9·9-s + 2.93·11-s − 19.0·13-s − 15·15-s + 122.·17-s − 107.·19-s − 21·21-s − 210.·23-s + 25·25-s − 27·27-s + 95.4·29-s + 94.3·31-s − 8.81·33-s + 35·35-s + 97.1·37-s + 57.1·39-s − 491.·41-s + 43.0·43-s + 45·45-s − 473.·47-s + 49·49-s − 367.·51-s − 183.·53-s + 14.6·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s + 0.377·7-s + 0.333·9-s + 0.0805·11-s − 0.406·13-s − 0.258·15-s + 1.74·17-s − 1.29·19-s − 0.218·21-s − 1.90·23-s + 0.200·25-s − 0.192·27-s + 0.611·29-s + 0.546·31-s − 0.0464·33-s + 0.169·35-s + 0.431·37-s + 0.234·39-s − 1.87·41-s + 0.152·43-s + 0.149·45-s − 1.46·47-s + 0.142·49-s − 1.00·51-s − 0.476·53-s + 0.0360·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 3T \) |
| 5 | \( 1 - 5T \) |
| 7 | \( 1 - 7T \) |
| good | 11 | \( 1 - 2.93T + 1.33e3T^{2} \) |
| 13 | \( 1 + 19.0T + 2.19e3T^{2} \) |
| 17 | \( 1 - 122.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 107.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 210.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 95.4T + 2.43e4T^{2} \) |
| 31 | \( 1 - 94.3T + 2.97e4T^{2} \) |
| 37 | \( 1 - 97.1T + 5.06e4T^{2} \) |
| 41 | \( 1 + 491.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 43.0T + 7.95e4T^{2} \) |
| 47 | \( 1 + 473.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 183.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 760.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 198.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 309.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 665.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 621.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 24.7T + 4.93e5T^{2} \) |
| 83 | \( 1 - 406.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 261.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.00e3T + 9.12e5T^{2} \) |
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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
−8.365873331690865021355234294023, −7.965628669218771443711135700562, −6.82254018549056952184892875912, −6.13295442324356444696925055783, −5.37020213120630974111840141763, −4.56279232557202726139426967714, −3.56151909403718124243672165284, −2.26767437987736129067043877648, −1.31435524573579029096390567214, 0,
1.31435524573579029096390567214, 2.26767437987736129067043877648, 3.56151909403718124243672165284, 4.56279232557202726139426967714, 5.37020213120630974111840141763, 6.13295442324356444696925055783, 6.82254018549056952184892875912, 7.965628669218771443711135700562, 8.365873331690865021355234294023
