| L(s) = 1 | + 1.79·3-s − 5·5-s + 7·7-s − 23.7·9-s + 32.2·11-s − 57.4·13-s − 8.95·15-s − 117.·17-s + 92.8·19-s + 12.5·21-s − 37.1·23-s + 25·25-s − 90.9·27-s + 254.·29-s + 202.·31-s + 57.7·33-s − 35·35-s + 58.1·37-s − 102.·39-s + 114.·41-s − 90.4·43-s + 118.·45-s + 273.·47-s + 49·49-s − 209.·51-s − 524.·53-s − 161.·55-s + ⋯ |
| L(s) = 1 | + 0.344·3-s − 0.447·5-s + 0.377·7-s − 0.881·9-s + 0.883·11-s − 1.22·13-s − 0.154·15-s − 1.67·17-s + 1.12·19-s + 0.130·21-s − 0.337·23-s + 0.200·25-s − 0.648·27-s + 1.62·29-s + 1.17·31-s + 0.304·33-s − 0.169·35-s + 0.258·37-s − 0.422·39-s + 0.436·41-s − 0.320·43-s + 0.394·45-s + 0.847·47-s + 0.142·49-s − 0.575·51-s − 1.36·53-s − 0.394·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1120 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.834615662\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.834615662\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + 5T \) |
| 7 | \( 1 - 7T \) |
| good | 3 | \( 1 - 1.79T + 27T^{2} \) |
| 11 | \( 1 - 32.2T + 1.33e3T^{2} \) |
| 13 | \( 1 + 57.4T + 2.19e3T^{2} \) |
| 17 | \( 1 + 117.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 92.8T + 6.85e3T^{2} \) |
| 23 | \( 1 + 37.1T + 1.21e4T^{2} \) |
| 29 | \( 1 - 254.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 202.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 58.1T + 5.06e4T^{2} \) |
| 41 | \( 1 - 114.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 90.4T + 7.95e4T^{2} \) |
| 47 | \( 1 - 273.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 524.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 601.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 141.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 565.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 676.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 28.7T + 3.89e5T^{2} \) |
| 79 | \( 1 + 309.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 370.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 325.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 979.T + 9.12e5T^{2} \) |
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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
−9.318443143058236882360766471751, −8.628850716272796374312746394731, −7.912971811132990587701850466942, −7.00553726706745106498633284335, −6.19609972656086473922367638086, −4.96555717245561156522584272655, −4.29506470806323418591219379078, −3.07176697426904151239776854383, −2.21480590826556521964788139768, −0.67767005772265348533043952480,
0.67767005772265348533043952480, 2.21480590826556521964788139768, 3.07176697426904151239776854383, 4.29506470806323418591219379078, 4.96555717245561156522584272655, 6.19609972656086473922367638086, 7.00553726706745106498633284335, 7.912971811132990587701850466942, 8.628850716272796374312746394731, 9.318443143058236882360766471751
