| L(s) = 1 | + 3·3-s + 19.1·5-s + 9·9-s + 40.5·11-s − 50.4·13-s + 57.4·15-s + 51.9·17-s + 33.1·19-s + 62.8·23-s + 241.·25-s + 27·27-s + 129.·29-s − 242.·31-s + 121.·33-s − 389.·37-s − 151.·39-s + 470.·41-s − 125.·43-s + 172.·45-s + 386.·47-s + 155.·51-s − 611.·53-s + 777.·55-s + 99.3·57-s − 226.·59-s − 725.·61-s − 966.·65-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.71·5-s + 0.333·9-s + 1.11·11-s − 1.07·13-s + 0.988·15-s + 0.740·17-s + 0.400·19-s + 0.569·23-s + 1.93·25-s + 0.192·27-s + 0.831·29-s − 1.40·31-s + 0.642·33-s − 1.73·37-s − 0.621·39-s + 1.79·41-s − 0.443·43-s + 0.570·45-s + 1.19·47-s + 0.427·51-s − 1.58·53-s + 1.90·55-s + 0.230·57-s − 0.499·59-s − 1.52·61-s − 1.84·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.678173226\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.678173226\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 3T \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 19.1T + 125T^{2} \) |
| 11 | \( 1 - 40.5T + 1.33e3T^{2} \) |
| 13 | \( 1 + 50.4T + 2.19e3T^{2} \) |
| 17 | \( 1 - 51.9T + 4.91e3T^{2} \) |
| 19 | \( 1 - 33.1T + 6.85e3T^{2} \) |
| 23 | \( 1 - 62.8T + 1.21e4T^{2} \) |
| 29 | \( 1 - 129.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 242.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 389.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 470.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 125.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 386.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 611.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 226.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 725.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 1.04e3T + 3.00e5T^{2} \) |
| 71 | \( 1 - 169.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 381.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.16e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 808.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 319.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.13e3T + 9.12e5T^{2} \) |
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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
−10.01766004911123719607660191921, −9.401452616915056795308262260820, −8.903228141809808767145617843228, −7.52685919045343486110822037873, −6.68251914610769123411660102763, −5.71590782311141448480120028254, −4.81352506681828629651156522151, −3.34685138088927157094285242409, −2.24050293296710511279718460527, −1.26774062652373213841259051934,
1.26774062652373213841259051934, 2.24050293296710511279718460527, 3.34685138088927157094285242409, 4.81352506681828629651156522151, 5.71590782311141448480120028254, 6.68251914610769123411660102763, 7.52685919045343486110822037873, 8.903228141809808767145617843228, 9.401452616915056795308262260820, 10.01766004911123719607660191921
