| L(s) = 1 | − 1.93i·2-s − 1.73·4-s + (−1.73 − 1.41i)5-s − 2.44i·7-s − 0.517i·8-s + (−2.73 + 3.34i)10-s + 4.73·11-s − 6.69i·13-s − 4.73·14-s − 4.46·16-s + 1.41i·17-s + 6.73·19-s + (3.00 + 2.44i)20-s − 9.14i·22-s − 3.48i·23-s + ⋯ |
| L(s) = 1 | − 1.36i·2-s − 0.866·4-s + (−0.774 − 0.632i)5-s − 0.925i·7-s − 0.183i·8-s + (−0.863 + 1.05i)10-s + 1.42·11-s − 1.85i·13-s − 1.26·14-s − 1.11·16-s + 0.342i·17-s + 1.54·19-s + (0.670 + 0.547i)20-s − 1.94i·22-s − 0.726i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1305 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.774 - 0.632i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1305 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.774 - 0.632i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.414035001\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.414035001\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + (1.73 + 1.41i)T \) |
| 29 | \( 1 - T \) |
| good | 2 | \( 1 + 1.93iT - 2T^{2} \) |
| 7 | \( 1 + 2.44iT - 7T^{2} \) |
| 11 | \( 1 - 4.73T + 11T^{2} \) |
| 13 | \( 1 + 6.69iT - 13T^{2} \) |
| 17 | \( 1 - 1.41iT - 17T^{2} \) |
| 19 | \( 1 - 6.73T + 19T^{2} \) |
| 23 | \( 1 + 3.48iT - 23T^{2} \) |
| 31 | \( 1 + 5.26T + 31T^{2} \) |
| 37 | \( 1 - 0.656iT - 37T^{2} \) |
| 41 | \( 1 + 6.92T + 41T^{2} \) |
| 43 | \( 1 + 0.656iT - 43T^{2} \) |
| 47 | \( 1 - 1.41iT - 47T^{2} \) |
| 53 | \( 1 - 8.76iT - 53T^{2} \) |
| 59 | \( 1 - 10.3T + 59T^{2} \) |
| 61 | \( 1 + 10.9T + 61T^{2} \) |
| 67 | \( 1 + 4.24iT - 67T^{2} \) |
| 71 | \( 1 - 3.46T + 71T^{2} \) |
| 73 | \( 1 - 7.34iT - 73T^{2} \) |
| 79 | \( 1 + 6.19T + 79T^{2} \) |
| 83 | \( 1 - 4.52iT - 83T^{2} \) |
| 89 | \( 1 + 10.3T + 89T^{2} \) |
| 97 | \( 1 - 6.03iT - 97T^{2} \) |
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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
−9.380860721469573057193298837810, −8.539235006832752637374099368818, −7.61638833275548819988407572603, −6.87804770674786852918725352498, −5.53363029759356559362906251168, −4.44578655545895883651390693756, −3.68976225311142910022509208897, −3.09204436440169087042401383369, −1.35534130192680599769842209377, −0.66200327723905549043118126433,
1.89511295743707659131701092410, 3.35889720522888311486638212939, 4.33378141021422295472863150773, 5.33530041851491305793208089580, 6.24941387113872404598489617600, 6.97585230613869630036786455512, 7.33707548855689254441456495251, 8.472544158681226764155610629729, 9.084350135220695634640221840927, 9.676718526599568965608621479433
