L(s) = 1 | + (0.195 + 0.980i)2-s + (1.38 − 1.38i)3-s + (−0.923 + 0.382i)4-s + (0.707 + 0.707i)5-s + (1.63 + 1.08i)6-s + (−0.555 − 0.831i)8-s − 2.84i·9-s + (−0.555 + 0.831i)10-s + (0.541 + 0.541i)11-s + (−0.750 + 1.81i)12-s + (−1.17 + 1.17i)13-s + 1.96·15-s + (0.707 − 0.707i)16-s + (2.79 − 0.555i)18-s + (0.707 − 0.707i)19-s + (−0.923 − 0.382i)20-s + ⋯ |
L(s) = 1 | + (0.195 + 0.980i)2-s + (1.38 − 1.38i)3-s + (−0.923 + 0.382i)4-s + (0.707 + 0.707i)5-s + (1.63 + 1.08i)6-s + (−0.555 − 0.831i)8-s − 2.84i·9-s + (−0.555 + 0.831i)10-s + (0.541 + 0.541i)11-s + (−0.750 + 1.81i)12-s + (−1.17 + 1.17i)13-s + 1.96·15-s + (0.707 − 0.707i)16-s + (2.79 − 0.555i)18-s + (0.707 − 0.707i)19-s + (−0.923 − 0.382i)20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 - 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 - 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.868785328\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.868785328\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.195 - 0.980i)T \) |
| 5 | \( 1 + (-0.707 - 0.707i)T \) |
| 19 | \( 1 + (-0.707 + 0.707i)T \) |
good | 3 | \( 1 + (-1.38 + 1.38i)T - iT^{2} \) |
| 7 | \( 1 + T^{2} \) |
| 11 | \( 1 + (-0.541 - 0.541i)T + iT^{2} \) |
| 13 | \( 1 + (1.17 - 1.17i)T - iT^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 23 | \( 1 + T^{2} \) |
| 29 | \( 1 + iT^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 + (0.275 + 0.275i)T + iT^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( 1 - iT^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 + (1.38 + 1.38i)T + iT^{2} \) |
| 59 | \( 1 - iT^{2} \) |
| 61 | \( 1 + (0.541 - 0.541i)T - iT^{2} \) |
| 67 | \( 1 + (0.785 - 0.785i)T - iT^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 + T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 + iT^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( 1 - 1.66T + T^{2} \) |
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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
−9.423141790621730040878596637329, −8.855692862300354798482937659867, −7.82236053371497753000696388304, −7.12669456532535078438428109661, −6.84639394924002370322252019475, −6.09633997072906205407500912915, −4.76157303712136607181817432326, −3.56161942812729849777783677692, −2.64614815788203547965382413550, −1.66953798124886658786403609740,
1.69972219851941271192881782158, 2.82333960686979936996818769627, 3.38716953950274229877931093051, 4.46741727891238614802913873201, 5.05401335729695709950397498642, 5.81604535537368864573379878158, 7.81662379637245830524077246488, 8.308307233691165577633226324194, 9.333122052703425606145644225081, 9.437878913920175219118714273158