L(s) = 1 | − 3.32·3-s + 2.74i·5-s − 4.32i·7-s + 8.07·9-s − i·11-s + (−3.60 − 0.140i)13-s − 9.12i·15-s + 2.28·17-s + 5.78i·19-s + 14.3i·21-s + 1.58·23-s − 2.52·25-s − 16.8·27-s − 1.44·29-s + 8.74i·31-s + ⋯ |
L(s) = 1 | − 1.92·3-s + 1.22i·5-s − 1.63i·7-s + 2.69·9-s − 0.301i·11-s + (−0.999 − 0.0390i)13-s − 2.35i·15-s + 0.553·17-s + 1.32i·19-s + 3.14i·21-s + 0.330·23-s − 0.505·25-s − 3.24·27-s − 0.269·29-s + 1.57i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 572 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0390 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 572 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0390 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.391142 + 0.376173i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.391142 + 0.376173i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 11 | \( 1 + iT \) |
| 13 | \( 1 + (3.60 + 0.140i)T \) |
good | 3 | \( 1 + 3.32T + 3T^{2} \) |
| 5 | \( 1 - 2.74iT - 5T^{2} \) |
| 7 | \( 1 + 4.32iT - 7T^{2} \) |
| 17 | \( 1 - 2.28T + 17T^{2} \) |
| 19 | \( 1 - 5.78iT - 19T^{2} \) |
| 23 | \( 1 - 1.58T + 23T^{2} \) |
| 29 | \( 1 + 1.44T + 29T^{2} \) |
| 31 | \( 1 - 8.74iT - 31T^{2} \) |
| 37 | \( 1 - 7.39iT - 37T^{2} \) |
| 41 | \( 1 - 4.33iT - 41T^{2} \) |
| 43 | \( 1 + 3.44T + 43T^{2} \) |
| 47 | \( 1 - 2.65iT - 47T^{2} \) |
| 53 | \( 1 - 11.8T + 53T^{2} \) |
| 59 | \( 1 - 0.986iT - 59T^{2} \) |
| 61 | \( 1 + 5.44T + 61T^{2} \) |
| 67 | \( 1 + 0.180iT - 67T^{2} \) |
| 71 | \( 1 - 15.1iT - 71T^{2} \) |
| 73 | \( 1 - 6.72iT - 73T^{2} \) |
| 79 | \( 1 - 12.6T + 79T^{2} \) |
| 83 | \( 1 + 11.5iT - 83T^{2} \) |
| 89 | \( 1 + 6.56iT - 89T^{2} \) |
| 97 | \( 1 - 5.25iT - 97T^{2} \) |
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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
−10.83102270874798983477269884846, −10.24202278598291193275352991887, −9.990509787064442955871382270883, −7.77695641179923893274000784291, −7.03242864714680645597307191413, −6.58460433767050360287999013564, −5.50551816034581986911931971407, −4.50503015871190510782457207083, −3.43024853924169661377162810687, −1.19924437337209641535381475414,
0.45659696260437373711446877668, 2.11485173961056099699713367907, 4.40346387346905002073277618754, 5.27658746744576742797012116251, 5.50923978957009018022002907142, 6.64888939083563307422546427163, 7.70830087589164313563825106076, 9.109641560957495118621290943854, 9.534119188524100486596843129087, 10.71395949996138924858541104553