L(s) = 1 | + (−0.366 + 1.36i)2-s + (−1.73 − i)4-s − 2.38·5-s + i·7-s + (2 − 1.99i)8-s + (0.873 − 3.25i)10-s − 1.65i·11-s + 0.253i·13-s + (−1.36 − 0.366i)14-s + (1.99 + 3.46i)16-s − 2i·17-s + 7.03·19-s + (4.13 + 2.38i)20-s + (2.25 + 0.605i)22-s − 2.82·23-s + ⋯ |
L(s) = 1 | + (−0.258 + 0.965i)2-s + (−0.866 − 0.5i)4-s − 1.06·5-s + 0.377i·7-s + (0.707 − 0.707i)8-s + (0.276 − 1.03i)10-s − 0.498i·11-s + 0.0702i·13-s + (−0.365 − 0.0978i)14-s + (0.499 + 0.866i)16-s − 0.485i·17-s + 1.61·19-s + (0.924 + 0.533i)20-s + (0.481 + 0.129i)22-s − 0.589·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7888209328\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7888209328\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.366 - 1.36i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 - iT \) |
good | 5 | \( 1 + 2.38T + 5T^{2} \) |
| 11 | \( 1 + 1.65iT - 11T^{2} \) |
| 13 | \( 1 - 0.253iT - 13T^{2} \) |
| 17 | \( 1 + 2iT - 17T^{2} \) |
| 19 | \( 1 - 7.03T + 19T^{2} \) |
| 23 | \( 1 + 2.82T + 23T^{2} \) |
| 29 | \( 1 - 1.26T + 29T^{2} \) |
| 31 | \( 1 - 0.746iT - 31T^{2} \) |
| 37 | \( 1 - 6.94iT - 37T^{2} \) |
| 41 | \( 1 - 5.55iT - 41T^{2} \) |
| 43 | \( 1 - 8.67T + 43T^{2} \) |
| 47 | \( 1 + 10.9T + 47T^{2} \) |
| 53 | \( 1 - 3.30T + 53T^{2} \) |
| 59 | \( 1 - 10.0iT - 59T^{2} \) |
| 61 | \( 1 - 4.53iT - 61T^{2} \) |
| 67 | \( 1 - 0.253T + 67T^{2} \) |
| 71 | \( 1 + 3.17T + 71T^{2} \) |
| 73 | \( 1 + 3.05T + 73T^{2} \) |
| 79 | \( 1 - 14.5iT - 79T^{2} \) |
| 83 | \( 1 - 1.77iT - 83T^{2} \) |
| 89 | \( 1 + 3.13iT - 89T^{2} \) |
| 97 | \( 1 + 5.49T + 97T^{2} \) |
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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
−9.566720071750369421773272075143, −8.793532021225186416212477810837, −7.998321829785879393578364600099, −7.54447951951849045471810191919, −6.64332916086923949118713456241, −5.73294214692846410574926652318, −4.92812604474218953220898966620, −4.00060270718038701052191088613, −3.01351308901792336976125411281, −1.04778429195741132066409832680,
0.43600860154425637532870647299, 1.81863035031247102616650201748, 3.16401719602875626703719783144, 3.88422582195439840615999400029, 4.62897243643960907455439054600, 5.67777023639755592181944480810, 7.15346083773938970348316426564, 7.70984655935349723527661192335, 8.365781241901194704292698997402, 9.376440220903453981529331502021