L(s) = 1 | + 0.162i·3-s − 1.05·5-s − i·7-s + 2.97·9-s + 6.18i·11-s − 5.55i·13-s − 0.171i·15-s − 3.90i·17-s + 2.97i·19-s + 0.162·21-s + 7.17·23-s − 3.87·25-s + 0.968i·27-s + 2.56i·29-s + 0.343·31-s + ⋯ |
L(s) = 1 | + 0.0935i·3-s − 0.473·5-s − 0.377i·7-s + 0.991·9-s + 1.86i·11-s − 1.53i·13-s − 0.0443i·15-s − 0.947i·17-s + 0.681i·19-s + 0.0353·21-s + 1.49·23-s − 0.775·25-s + 0.186i·27-s + 0.475i·29-s + 0.0616·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 - 0.110i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.993 - 0.110i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.624865208\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.624865208\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 + iT \) |
| 41 | \( 1 + (-0.710 - 6.36i)T \) |
good | 3 | \( 1 - 0.162iT - 3T^{2} \) |
| 5 | \( 1 + 1.05T + 5T^{2} \) |
| 11 | \( 1 - 6.18iT - 11T^{2} \) |
| 13 | \( 1 + 5.55iT - 13T^{2} \) |
| 17 | \( 1 + 3.90iT - 17T^{2} \) |
| 19 | \( 1 - 2.97iT - 19T^{2} \) |
| 23 | \( 1 - 7.17T + 23T^{2} \) |
| 29 | \( 1 - 2.56iT - 29T^{2} \) |
| 31 | \( 1 - 0.343T + 31T^{2} \) |
| 37 | \( 1 - 8.19T + 37T^{2} \) |
| 43 | \( 1 - 10.0T + 43T^{2} \) |
| 47 | \( 1 + 7.52iT - 47T^{2} \) |
| 53 | \( 1 - 1.52iT - 53T^{2} \) |
| 59 | \( 1 - 11.7T + 59T^{2} \) |
| 61 | \( 1 - 11.1T + 61T^{2} \) |
| 67 | \( 1 + 4.12iT - 67T^{2} \) |
| 71 | \( 1 - 7.66iT - 71T^{2} \) |
| 73 | \( 1 + 12.2T + 73T^{2} \) |
| 79 | \( 1 + 11.8iT - 79T^{2} \) |
| 83 | \( 1 - 14.0T + 83T^{2} \) |
| 89 | \( 1 - 1.32iT - 89T^{2} \) |
| 97 | \( 1 + 1.93iT - 97T^{2} \) |
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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
−9.962399175988468115474398985851, −9.182853955023268668055313816607, −7.81940595033776995300879367920, −7.48185949996823759944280012539, −6.74096889734917518620104559050, −5.34745257442012645069869754312, −4.60098437537828334117947342760, −3.76095853140355128236519614765, −2.51092331749428022595212674579, −1.03413514005359502483381782368,
0.994936003286524678096053701278, 2.44349618805520341059300439117, 3.73423877647959522413399755975, 4.39452327896842654765137465466, 5.66010552598103782079570154993, 6.46546833820368254246464535957, 7.27600506498796629626773923310, 8.232706658601049991337170796711, 8.952718934386217209663423879687, 9.603237800201735451605357451712