L(s) = 1 | − i·2-s + (1 − i)3-s − 4-s + (2 − i)5-s + (−1 − i)6-s − 2·7-s + i·8-s + i·9-s + (−1 − 2i)10-s + (−1 + i)11-s + (−1 + i)12-s + (2 − 3i)13-s + 2i·14-s + (1 − 3i)15-s + 16-s + (−1 + i)17-s + ⋯ |
L(s) = 1 | − 0.707i·2-s + (0.577 − 0.577i)3-s − 0.5·4-s + (0.894 − 0.447i)5-s + (−0.408 − 0.408i)6-s − 0.755·7-s + 0.353i·8-s + 0.333i·9-s + (−0.316 − 0.632i)10-s + (−0.301 + 0.301i)11-s + (−0.288 + 0.288i)12-s + (0.554 − 0.832i)13-s + 0.534i·14-s + (0.258 − 0.774i)15-s + 0.250·16-s + (−0.242 + 0.242i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 130 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.134 + 0.990i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 130 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.134 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.954349 - 0.833303i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.954349 - 0.833303i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + iT \) |
| 5 | \( 1 + (-2 + i)T \) |
| 13 | \( 1 + (-2 + 3i)T \) |
good | 3 | \( 1 + (-1 + i)T - 3iT^{2} \) |
| 7 | \( 1 + 2T + 7T^{2} \) |
| 11 | \( 1 + (1 - i)T - 11iT^{2} \) |
| 17 | \( 1 + (1 - i)T - 17iT^{2} \) |
| 19 | \( 1 + (3 - 3i)T - 19iT^{2} \) |
| 23 | \( 1 + (-1 - i)T + 23iT^{2} \) |
| 29 | \( 1 - 8iT - 29T^{2} \) |
| 31 | \( 1 + (-1 - i)T + 31iT^{2} \) |
| 37 | \( 1 - 8T + 37T^{2} \) |
| 41 | \( 1 + (7 + 7i)T + 41iT^{2} \) |
| 43 | \( 1 + (-1 - i)T + 43iT^{2} \) |
| 47 | \( 1 + 10T + 47T^{2} \) |
| 53 | \( 1 + (-1 + i)T - 53iT^{2} \) |
| 59 | \( 1 + (9 + 9i)T + 59iT^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 + 12iT - 67T^{2} \) |
| 71 | \( 1 + (-5 - 5i)T + 71iT^{2} \) |
| 73 | \( 1 + 6iT - 73T^{2} \) |
| 79 | \( 1 + 10iT - 79T^{2} \) |
| 83 | \( 1 + 18T + 83T^{2} \) |
| 89 | \( 1 + (-11 - 11i)T + 89iT^{2} \) |
| 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
−12.92358644331471854043348906361, −12.59885894662536654305191701361, −10.82998032523111201228315194521, −10.03163407259311614154419274879, −8.925371997439647766362272555654, −8.002957984633436240618410672943, −6.42182849287379615300516084221, −5.07529002721091630239953100297, −3.17974475417872030817262495352, −1.78487513909741876656119990510,
2.88718171924783037467530224203, 4.36437902718636375213579142766, 6.08369063518413757126183913516, 6.74498304168338119841727422465, 8.436428541187349539410498002341, 9.396034766172495879402545659133, 9.988714052225930792873778562481, 11.35548037414773873533430140691, 13.05459281641719259178489730271, 13.64337110530095432106357131999