| L(s) = 1 | − 3.37·2-s + 3.40·4-s + 15.7·5-s − 17.1·7-s + 15.5·8-s − 53.0·10-s − 52.8·11-s + 57.9·14-s − 79.6·16-s + 71.0·17-s + 92.6·19-s + 53.5·20-s + 178.·22-s − 190.·23-s + 121.·25-s − 58.4·28-s + 128.·29-s + 3.29·31-s + 144.·32-s − 240.·34-s − 269.·35-s + 241.·37-s − 313.·38-s + 243.·40-s + 97.1·41-s + 376.·43-s − 179.·44-s + ⋯ |
| L(s) = 1 | − 1.19·2-s + 0.425·4-s + 1.40·5-s − 0.926·7-s + 0.685·8-s − 1.67·10-s − 1.44·11-s + 1.10·14-s − 1.24·16-s + 1.01·17-s + 1.11·19-s + 0.598·20-s + 1.72·22-s − 1.72·23-s + 0.975·25-s − 0.394·28-s + 0.820·29-s + 0.0191·31-s + 0.800·32-s − 1.21·34-s − 1.30·35-s + 1.07·37-s − 1.33·38-s + 0.963·40-s + 0.370·41-s + 1.33·43-s − 0.616·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 3.37T + 8T^{2} \) |
| 5 | \( 1 - 15.7T + 125T^{2} \) |
| 7 | \( 1 + 17.1T + 343T^{2} \) |
| 11 | \( 1 + 52.8T + 1.33e3T^{2} \) |
| 17 | \( 1 - 71.0T + 4.91e3T^{2} \) |
| 19 | \( 1 - 92.6T + 6.85e3T^{2} \) |
| 23 | \( 1 + 190.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 128.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 3.29T + 2.97e4T^{2} \) |
| 37 | \( 1 - 241.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 97.1T + 6.89e4T^{2} \) |
| 43 | \( 1 - 376.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 577.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 307.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 349.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 127.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 903.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 826.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 131.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 556.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 254.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 183.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 780.T + 9.12e5T^{2} \) |
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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
−8.901668504346623491368701031486, −7.908821094896134548913154302338, −7.43274038771469940108879802286, −6.18040273809091709290712900081, −5.69558794078248078552984562199, −4.65655762704620411333653027751, −3.12327290638370777138077104626, −2.25624462383218863338117684689, −1.15918264399876418770142977721, 0,
1.15918264399876418770142977721, 2.25624462383218863338117684689, 3.12327290638370777138077104626, 4.65655762704620411333653027751, 5.69558794078248078552984562199, 6.18040273809091709290712900081, 7.43274038771469940108879802286, 7.908821094896134548913154302338, 8.901668504346623491368701031486
