| L(s) = 1 | + (−4.79 + 1.41i)5-s − 6.78i·7-s − 9.89i·11-s − 20.3i·13-s − 19.1·17-s + 12·19-s − 9.59·23-s + (20.9 − 13.5i)25-s + 8.48i·29-s − 38·31-s + (9.59 + 32.5i)35-s − 6.78i·37-s − 69.2i·41-s + 67.8i·43-s + 76.7·47-s + ⋯ |
| L(s) = 1 | + (−0.959 + 0.282i)5-s − 0.968i·7-s − 0.899i·11-s − 1.56i·13-s − 1.12·17-s + 0.631·19-s − 0.417·23-s + (0.839 − 0.542i)25-s + 0.292i·29-s − 1.22·31-s + (0.274 + 0.929i)35-s − 0.183i·37-s − 1.69i·41-s + 1.57i·43-s + 1.63·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.322 + 0.946i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.322 + 0.946i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.509437 - 0.712027i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.509437 - 0.712027i\) |
| \(L(2)\) |
|
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 | \( 1 + (4.79 - 1.41i)T \) |
| good | 7 | \( 1 + 6.78iT - 49T^{2} \) |
| 11 | \( 1 + 9.89iT - 121T^{2} \) |
| 13 | \( 1 + 20.3iT - 169T^{2} \) |
| 17 | \( 1 + 19.1T + 289T^{2} \) |
| 19 | \( 1 - 12T + 361T^{2} \) |
| 23 | \( 1 + 9.59T + 529T^{2} \) |
| 29 | \( 1 - 8.48iT - 841T^{2} \) |
| 31 | \( 1 + 38T + 961T^{2} \) |
| 37 | \( 1 + 6.78iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 69.2iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 67.8iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 76.7T + 2.20e3T^{2} \) |
| 53 | \( 1 + 2.80e3T^{2} \) |
| 59 | \( 1 - 83.4iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 70T + 3.72e3T^{2} \) |
| 67 | \( 1 - 108. iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 118. iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 13.5iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 30T + 6.24e3T^{2} \) |
| 83 | \( 1 - 134.T + 6.88e3T^{2} \) |
| 89 | \( 1 + 32.5iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 94.9iT - 9.40e3T^{2} \) |
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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
−12.09059275937874581262243414925, −10.88048790785981361084278375638, −10.58339043215470796643680255113, −8.963344989416411720969739558536, −7.86832162683763474744564647009, −7.14752593323070181919724730303, −5.69911033546746498335543437268, −4.17556390043276308133787899170, −3.12459221427621434247437032521, −0.51887566594383282620288571441,
2.11809332758381935180265458143, 3.97170331071021253527328506416, 5.00358445468564381397862763256, 6.55945702643804471634541379027, 7.57886488420721303517183902323, 8.829667365772288083047372409978, 9.436033134814924322075518487872, 11.01834157280556348993118073184, 11.87335879295262237337597121847, 12.41820047429628731353447077646
