| L(s) = 1 | − 5·5-s + 24.6·7-s + 62.6·11-s + 59.6·13-s + 85.2·17-s − 75.1·19-s + 97.5·23-s + 25·25-s − 159.·29-s + 318.·31-s − 123.·35-s + 393.·37-s − 85.0·41-s − 40.7·43-s + 90.2·47-s + 262.·49-s − 691.·53-s − 313.·55-s − 283.·59-s − 432.·61-s − 298.·65-s + 308.·67-s − 302.·71-s − 457.·73-s + 1.54e3·77-s − 841.·79-s + 788.·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 1.32·7-s + 1.71·11-s + 1.27·13-s + 1.21·17-s − 0.907·19-s + 0.884·23-s + 0.200·25-s − 1.01·29-s + 1.84·31-s − 0.594·35-s + 1.74·37-s − 0.324·41-s − 0.144·43-s + 0.280·47-s + 0.764·49-s − 1.79·53-s − 0.767·55-s − 0.625·59-s − 0.907·61-s − 0.569·65-s + 0.562·67-s − 0.505·71-s − 0.733·73-s + 2.28·77-s − 1.19·79-s + 1.04·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.276024155\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.276024155\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + 5T \) |
| good | 7 | \( 1 - 24.6T + 343T^{2} \) |
| 11 | \( 1 - 62.6T + 1.33e3T^{2} \) |
| 13 | \( 1 - 59.6T + 2.19e3T^{2} \) |
| 17 | \( 1 - 85.2T + 4.91e3T^{2} \) |
| 19 | \( 1 + 75.1T + 6.85e3T^{2} \) |
| 23 | \( 1 - 97.5T + 1.21e4T^{2} \) |
| 29 | \( 1 + 159.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 318.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 393.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 85.0T + 6.89e4T^{2} \) |
| 43 | \( 1 + 40.7T + 7.95e4T^{2} \) |
| 47 | \( 1 - 90.2T + 1.03e5T^{2} \) |
| 53 | \( 1 + 691.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 283.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 432.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 308.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 302.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 457.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 841.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 788.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.11e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.19e3T + 9.12e5T^{2} \) |
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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
−8.876239896109957749123577998977, −8.253244225801462976690129349044, −7.61075691782560908910632718402, −6.53360699024634576973102063085, −5.89796385029119922604983070847, −4.66939704816952830553052333402, −4.10490707319342207677877877499, −3.13750902029127876448362537871, −1.56885345394106062950242749786, −1.01570011958784443556124629417,
1.01570011958784443556124629417, 1.56885345394106062950242749786, 3.13750902029127876448362537871, 4.10490707319342207677877877499, 4.66939704816952830553052333402, 5.89796385029119922604983070847, 6.53360699024634576973102063085, 7.61075691782560908910632718402, 8.253244225801462976690129349044, 8.876239896109957749123577998977
