| L(s) = 1 | − i·2-s + 3-s − 4-s + 3i·5-s − i·6-s − 2i·7-s + i·8-s + 9-s + 3·10-s − 6i·11-s − 12-s − 2·14-s + 3i·15-s + 16-s + 3·17-s − i·18-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s + 0.577·3-s − 0.5·4-s + 1.34i·5-s − 0.408i·6-s − 0.755i·7-s + 0.353i·8-s + 0.333·9-s + 0.948·10-s − 1.80i·11-s − 0.288·12-s − 0.534·14-s + 0.774i·15-s + 0.250·16-s + 0.727·17-s − 0.235i·18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.554 + 0.832i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.554 + 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.914219842\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.914219842\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + iT \) |
| 3 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 - 3iT - 5T^{2} \) |
| 7 | \( 1 + 2iT - 7T^{2} \) |
| 11 | \( 1 + 6iT - 11T^{2} \) |
| 17 | \( 1 - 3T + 17T^{2} \) |
| 19 | \( 1 - 2iT - 19T^{2} \) |
| 23 | \( 1 - 6T + 23T^{2} \) |
| 29 | \( 1 - 3T + 29T^{2} \) |
| 31 | \( 1 + 4iT - 31T^{2} \) |
| 37 | \( 1 - 7iT - 37T^{2} \) |
| 41 | \( 1 + 3iT - 41T^{2} \) |
| 43 | \( 1 - 10T + 43T^{2} \) |
| 47 | \( 1 + 6iT - 47T^{2} \) |
| 53 | \( 1 - 3T + 53T^{2} \) |
| 59 | \( 1 - 59T^{2} \) |
| 61 | \( 1 + 7T + 61T^{2} \) |
| 67 | \( 1 + 10iT - 67T^{2} \) |
| 71 | \( 1 - 6iT - 71T^{2} \) |
| 73 | \( 1 - 13iT - 73T^{2} \) |
| 79 | \( 1 + 4T + 79T^{2} \) |
| 83 | \( 1 + 6iT - 83T^{2} \) |
| 89 | \( 1 + 18iT - 89T^{2} \) |
| 97 | \( 1 - 14iT - 97T^{2} \) |
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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
−10.09694217243966762563634774920, −9.078722535519452308386621994420, −8.219978263515380867421639487113, −7.42479235173276700978165342461, −6.51585432110654090064465268534, −5.52830597670793300481902384933, −4.04204691639570562154107595401, −3.27399901836417283259677689636, −2.70202374681723769908966333305, −1.00201846058436211373320105939,
1.30260739923111111729756136444, 2.66465392061265434602738065832, 4.23834804277256824000914338209, 4.88452264775728862718748791489, 5.62108289814491207044824502621, 6.92275062589676140230407220281, 7.62238022081785491445680933340, 8.491109486648388807856001787039, 9.255335132039047088910702636810, 9.498289266918123704869803459900
