L(s) = 1 | − 2.51i·2-s − i·3-s − 4.30·4-s + (−1.80 − 1.31i)5-s − 2.51·6-s + i·7-s + 5.77i·8-s − 9-s + (−3.30 + 4.53i)10-s − 11-s + 4.30i·12-s + 7.07i·13-s + 2.51·14-s + (−1.31 + 1.80i)15-s + 5.89·16-s − 0.968i·17-s + ⋯ |
L(s) = 1 | − 1.77i·2-s − 0.577i·3-s − 2.15·4-s + (−0.807 − 0.589i)5-s − 1.02·6-s + 0.377i·7-s + 2.04i·8-s − 0.333·9-s + (−1.04 + 1.43i)10-s − 0.301·11-s + 1.24i·12-s + 1.96i·13-s + 0.670·14-s + (−0.340 + 0.466i)15-s + 1.47·16-s − 0.234i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.807 + 0.589i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.807 + 0.589i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4901691836\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4901691836\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + iT \) |
| 5 | \( 1 + (1.80 + 1.31i)T \) |
| 7 | \( 1 - iT \) |
| 11 | \( 1 + T \) |
good | 2 | \( 1 + 2.51iT - 2T^{2} \) |
| 13 | \( 1 - 7.07iT - 13T^{2} \) |
| 17 | \( 1 + 0.968iT - 17T^{2} \) |
| 19 | \( 1 - 2.31T + 19T^{2} \) |
| 23 | \( 1 + 5.12iT - 23T^{2} \) |
| 29 | \( 1 + 6.98T + 29T^{2} \) |
| 31 | \( 1 - 6.96T + 31T^{2} \) |
| 37 | \( 1 + 4.39iT - 37T^{2} \) |
| 41 | \( 1 + 8.89T + 41T^{2} \) |
| 43 | \( 1 - 12.7iT - 43T^{2} \) |
| 47 | \( 1 - 7.09iT - 47T^{2} \) |
| 53 | \( 1 - 5.21iT - 53T^{2} \) |
| 59 | \( 1 + 8.48T + 59T^{2} \) |
| 61 | \( 1 + 1.53T + 61T^{2} \) |
| 67 | \( 1 + 3.70iT - 67T^{2} \) |
| 71 | \( 1 - 11.7T + 71T^{2} \) |
| 73 | \( 1 - 1.94iT - 73T^{2} \) |
| 79 | \( 1 + 13.1T + 79T^{2} \) |
| 83 | \( 1 + 8.06iT - 83T^{2} \) |
| 89 | \( 1 - 12.6T + 89T^{2} \) |
| 97 | \( 1 + 4.07iT - 97T^{2} \) |
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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
−9.639653030088523072812047718610, −9.122638611997092462935237017228, −8.418802915485619064075448725775, −7.47805082499123133592642690672, −6.32285946966414950574253869556, −4.88595998452500852120482766587, −4.34693858247374175397089504420, −3.26576980625663585557170726308, −2.21480344832851838642909898581, −1.22348841234654297872399780261,
0.24118196483591737243992979579, 3.19517321745655297601057489676, 3.90208316661413577899396465157, 5.11069158825390534784820277298, 5.57657104196766965844901045719, 6.67253933872921234640520718239, 7.47861515817667052244449074933, 8.007157431917367863213581000995, 8.622784666708842267822075784494, 9.840774069974778193400827536046