L(s) = 1 | + (0.707 − 0.707i)2-s + (0.267 + 0.267i)3-s − 1.00i·4-s + 0.378·6-s + (−0.709 − 0.709i)7-s + (−0.707 − 0.707i)8-s − 2.85i·9-s − 5.59·11-s + (0.267 − 0.267i)12-s + (−2.99 − 2.99i)13-s − 1.00·14-s − 1.00·16-s + (2.20 + 2.20i)17-s + (−2.02 − 2.02i)18-s + (3.84 − 2.05i)19-s + ⋯ |
L(s) = 1 | + (0.499 − 0.499i)2-s + (0.154 + 0.154i)3-s − 0.500i·4-s + 0.154·6-s + (−0.268 − 0.268i)7-s + (−0.250 − 0.250i)8-s − 0.952i·9-s − 1.68·11-s + (0.0772 − 0.0772i)12-s + (−0.830 − 0.830i)13-s − 0.268·14-s − 0.250·16-s + (0.534 + 0.534i)17-s + (−0.476 − 0.476i)18-s + (0.881 − 0.472i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.935 + 0.353i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.935 + 0.353i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.207610 - 1.13822i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.207610 - 1.13822i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.707 + 0.707i)T \) |
| 5 | \( 1 \) |
| 19 | \( 1 + (-3.84 + 2.05i)T \) |
good | 3 | \( 1 + (-0.267 - 0.267i)T + 3iT^{2} \) |
| 7 | \( 1 + (0.709 + 0.709i)T + 7iT^{2} \) |
| 11 | \( 1 + 5.59T + 11T^{2} \) |
| 13 | \( 1 + (2.99 + 2.99i)T + 13iT^{2} \) |
| 17 | \( 1 + (-2.20 - 2.20i)T + 17iT^{2} \) |
| 23 | \( 1 + (5.89 - 5.89i)T - 23iT^{2} \) |
| 29 | \( 1 + 9.34T + 29T^{2} \) |
| 31 | \( 1 + 3.46iT - 31T^{2} \) |
| 37 | \( 1 + (-6.61 + 6.61i)T - 37iT^{2} \) |
| 41 | \( 1 + 9.14iT - 41T^{2} \) |
| 43 | \( 1 + (-0.927 + 0.927i)T - 43iT^{2} \) |
| 47 | \( 1 + (-4.20 - 4.20i)T + 47iT^{2} \) |
| 53 | \( 1 + (-2.44 - 2.44i)T + 53iT^{2} \) |
| 59 | \( 1 - 12.5T + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 + (5.72 - 5.72i)T - 67iT^{2} \) |
| 71 | \( 1 + 13.9iT - 71T^{2} \) |
| 73 | \( 1 + (4.33 - 4.33i)T - 73iT^{2} \) |
| 79 | \( 1 + 5.47T + 79T^{2} \) |
| 83 | \( 1 + (5.53 - 5.53i)T - 83iT^{2} \) |
| 89 | \( 1 - 8.99T + 89T^{2} \) |
| 97 | \( 1 + (-9.16 + 9.16i)T - 97iT^{2} \) |
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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
−9.896655811018227751011855684535, −9.129856045945488465243869312159, −7.74997255289080728624710330746, −7.35756339064743344962963184253, −5.70117612075482033817494185662, −5.51573939149571849177775815950, −4.06205239204749792587477987728, −3.28225195506269379055227330345, −2.27388651927320344411989253166, −0.41202892990076630357173821014,
2.20430890534239956773400310077, 2.98973318844871366394810620913, 4.46190673260912376350235537686, 5.21129055274863947927784581565, 5.95956791718817679294796683278, 7.22073042246452423529182111507, 7.73133685496596656417317189609, 8.432867779069236427421997876294, 9.674497662782136920513663974057, 10.25921351843344649715680381588