L(s) = 1 | + 0.755i·3-s − 172. i·7-s + 242.·9-s − 391.·11-s + 149. i·13-s − 1.18e3i·17-s + 685.·19-s + 130.·21-s − 996. i·23-s + 366. i·27-s + 8.76e3·29-s − 9.52e3·31-s − 295. i·33-s − 1.02e4i·37-s − 112.·39-s + ⋯ |
L(s) = 1 | + 0.0484i·3-s − 1.33i·7-s + 0.997·9-s − 0.975·11-s + 0.244i·13-s − 0.996i·17-s + 0.435·19-s + 0.0644·21-s − 0.392i·23-s + 0.0968i·27-s + 1.93·29-s − 1.78·31-s − 0.0472i·33-s − 1.22i·37-s − 0.0118·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(1.315595240\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.315595240\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
good | 3 | \( 1 - 0.755iT - 243T^{2} \) |
| 7 | \( 1 + 172. iT - 1.68e4T^{2} \) |
| 11 | \( 1 + 391.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 149. iT - 3.71e5T^{2} \) |
| 17 | \( 1 + 1.18e3iT - 1.41e6T^{2} \) |
| 19 | \( 1 - 685.T + 2.47e6T^{2} \) |
| 23 | \( 1 + 996. iT - 6.43e6T^{2} \) |
| 29 | \( 1 - 8.76e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 9.52e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.02e4iT - 6.93e7T^{2} \) |
| 41 | \( 1 - 32.6T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.03e4iT - 1.47e8T^{2} \) |
| 47 | \( 1 - 1.69e4iT - 2.29e8T^{2} \) |
| 53 | \( 1 + 2.22e4iT - 4.18e8T^{2} \) |
| 59 | \( 1 - 4.42e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 2.17e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 2.07e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 + 1.53e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 5.79e4iT - 2.07e9T^{2} \) |
| 79 | \( 1 + 6.24e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 4.35e4iT - 3.93e9T^{2} \) |
| 89 | \( 1 + 6.62e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.22e4iT - 8.58e9T^{2} \) |
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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
−9.311519342706840130389196881269, −8.154080101409346909197454274523, −7.20900424848506809401022529105, −6.95802845569915738881736842882, −5.46150561942609743568482034057, −4.56692722171304283951590048245, −3.78141817063589530651375721245, −2.55442155886327519466000580246, −1.21357507274859323491321590128, −0.27681183389687099342744382260,
1.32286767428126394623184451968, 2.35116959640712297916446654181, 3.35198040128641101194588894883, 4.65554303200799932143016424426, 5.47312472797462897962350718093, 6.30929240576507647884570538603, 7.38673179743830816769745115859, 8.220178361743954247709723867857, 8.957627124361555033148118428810, 9.968532695775845427632545645869