L(s) = 1 | − 2-s + (−0.794 + 0.794i)3-s + 4-s + (2.22 − 0.217i)5-s + (0.794 − 0.794i)6-s + (2.93 − 2.93i)7-s − 8-s + 1.73i·9-s + (−2.22 + 0.217i)10-s − 3.55i·11-s + (−0.794 + 0.794i)12-s − 4.41·13-s + (−2.93 + 2.93i)14-s + (−1.59 + 1.94i)15-s + 16-s − 5.37i·17-s + ⋯ |
L(s) = 1 | − 0.707·2-s + (−0.458 + 0.458i)3-s + 0.5·4-s + (0.995 − 0.0971i)5-s + (0.324 − 0.324i)6-s + (1.11 − 1.11i)7-s − 0.353·8-s + 0.578i·9-s + (−0.703 + 0.0686i)10-s − 1.07i·11-s + (−0.229 + 0.229i)12-s − 1.22·13-s + (−0.785 + 0.785i)14-s + (−0.412 + 0.501i)15-s + 0.250·16-s − 1.30i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.960 + 0.278i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.960 + 0.278i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.07203 - 0.152151i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.07203 - 0.152151i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 5 | \( 1 + (-2.22 + 0.217i)T \) |
| 37 | \( 1 + (3.88 - 4.67i)T \) |
good | 3 | \( 1 + (0.794 - 0.794i)T - 3iT^{2} \) |
| 7 | \( 1 + (-2.93 + 2.93i)T - 7iT^{2} \) |
| 11 | \( 1 + 3.55iT - 11T^{2} \) |
| 13 | \( 1 + 4.41T + 13T^{2} \) |
| 17 | \( 1 + 5.37iT - 17T^{2} \) |
| 19 | \( 1 + (-1.98 - 1.98i)T + 19iT^{2} \) |
| 23 | \( 1 - 5.90T + 23T^{2} \) |
| 29 | \( 1 + (-2.18 + 2.18i)T - 29iT^{2} \) |
| 31 | \( 1 + (-5.42 - 5.42i)T + 31iT^{2} \) |
| 41 | \( 1 - 6.34iT - 41T^{2} \) |
| 43 | \( 1 - 1.78T + 43T^{2} \) |
| 47 | \( 1 + (-1.82 + 1.82i)T - 47iT^{2} \) |
| 53 | \( 1 + (9.57 + 9.57i)T + 53iT^{2} \) |
| 59 | \( 1 + (-4.44 - 4.44i)T + 59iT^{2} \) |
| 61 | \( 1 + (-1.44 - 1.44i)T + 61iT^{2} \) |
| 67 | \( 1 + (-7.88 - 7.88i)T + 67iT^{2} \) |
| 71 | \( 1 + 11.2T + 71T^{2} \) |
| 73 | \( 1 + (4.02 - 4.02i)T - 73iT^{2} \) |
| 79 | \( 1 + (-2.03 - 2.03i)T + 79iT^{2} \) |
| 83 | \( 1 + (4.51 + 4.51i)T + 83iT^{2} \) |
| 89 | \( 1 + (-2.13 + 2.13i)T - 89iT^{2} \) |
| 97 | \( 1 - 5.91iT - 97T^{2} \) |
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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
−11.15977843870280306329852098597, −10.30497663616130942698089073243, −9.827548396598863810634408560008, −8.617667894641725297588222596649, −7.67071331267999864914369787297, −6.74805779448185809762994351184, −5.28909296845585937868484866205, −4.77753966138289939247282244965, −2.76775290076650559342957943992, −1.13243733653321496779901018892,
1.56838593941602592229876276462, 2.52880547176859144573578115747, 4.86125689953495117845036051287, 5.74449041787422537453894757237, 6.74714326627825856826900219914, 7.63950330732654339633570755720, 8.881845044047381395547531017892, 9.470754951881793384283145919485, 10.46776951170922528118666944345, 11.42964670132454060702834745774