L(s) = 1 | + 2-s + (−0.707 + 0.707i)3-s + 4-s + (−1.19 − 1.88i)5-s + (−0.707 + 0.707i)6-s + (−1.50 + 1.50i)7-s + 8-s − 1.00i·9-s + (−1.19 − 1.88i)10-s + 0.249i·11-s + (−0.707 + 0.707i)12-s − 2.83·13-s + (−1.50 + 1.50i)14-s + (2.18 + 0.486i)15-s + 16-s + 7.29i·17-s + ⋯ |
L(s) = 1 | + 0.707·2-s + (−0.408 + 0.408i)3-s + 0.5·4-s + (−0.536 − 0.843i)5-s + (−0.288 + 0.288i)6-s + (−0.569 + 0.569i)7-s + 0.353·8-s − 0.333i·9-s + (−0.379 − 0.596i)10-s + 0.0750i·11-s + (−0.204 + 0.204i)12-s − 0.785·13-s + (−0.402 + 0.402i)14-s + (0.563 + 0.125i)15-s + 0.250·16-s + 1.76i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.908 - 0.418i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.908 - 0.418i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5285471929\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5285471929\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 + (0.707 - 0.707i)T \) |
| 5 | \( 1 + (1.19 + 1.88i)T \) |
| 37 | \( 1 + (5.25 + 3.06i)T \) |
good | 7 | \( 1 + (1.50 - 1.50i)T - 7iT^{2} \) |
| 11 | \( 1 - 0.249iT - 11T^{2} \) |
| 13 | \( 1 + 2.83T + 13T^{2} \) |
| 17 | \( 1 - 7.29iT - 17T^{2} \) |
| 19 | \( 1 + (1.89 + 1.89i)T + 19iT^{2} \) |
| 23 | \( 1 + 8.28T + 23T^{2} \) |
| 29 | \( 1 + (-2.37 + 2.37i)T - 29iT^{2} \) |
| 31 | \( 1 + (-1.83 - 1.83i)T + 31iT^{2} \) |
| 41 | \( 1 - 10.3iT - 41T^{2} \) |
| 43 | \( 1 + 7.61T + 43T^{2} \) |
| 47 | \( 1 + (2.82 - 2.82i)T - 47iT^{2} \) |
| 53 | \( 1 + (-4.20 - 4.20i)T + 53iT^{2} \) |
| 59 | \( 1 + (6.26 + 6.26i)T + 59iT^{2} \) |
| 61 | \( 1 + (3.22 + 3.22i)T + 61iT^{2} \) |
| 67 | \( 1 + (5.68 + 5.68i)T + 67iT^{2} \) |
| 71 | \( 1 - 3.37T + 71T^{2} \) |
| 73 | \( 1 + (-7.48 + 7.48i)T - 73iT^{2} \) |
| 79 | \( 1 + (5.99 + 5.99i)T + 79iT^{2} \) |
| 83 | \( 1 + (-5.98 - 5.98i)T + 83iT^{2} \) |
| 89 | \( 1 + (5.37 - 5.37i)T - 89iT^{2} \) |
| 97 | \( 1 - 5.94iT - 97T^{2} \) |
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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
−10.21657299334284582525454405785, −9.501473334202754161524952802697, −8.469393308690393113779091274786, −7.81393585492638602317107493456, −6.46874557459171423447393100460, −5.93980854490115320640249965807, −4.89382853432751169688791431112, −4.24885866247071573059970983309, −3.30150821743869219524892109045, −1.86609970624145477096414467792,
0.17819005109592515270215963603, 2.21763358202485704684038973695, 3.23101288289643418961155574028, 4.16385622618908585918827092999, 5.18109890343947314071661299300, 6.21509266160821745159746593137, 7.01974269876168750921321666295, 7.37075979508312934553779623282, 8.422810840916270981274313002220, 9.965788339626651668700839855736