L(s) = 1 | + (−1.22 − 0.707i)5-s + (1.22 − 0.707i)11-s + (−0.5 + 0.866i)13-s − 1.41i·17-s − 19-s + (−1.22 − 0.707i)23-s + (0.499 + 0.866i)25-s + (0.5 − 0.866i)31-s + (−0.5 − 0.866i)43-s + (0.5 − 0.866i)49-s + 1.41i·53-s − 2·55-s + (−1.22 − 0.707i)59-s + (−0.5 − 0.866i)61-s + (1.22 − 0.707i)65-s + ⋯ |
L(s) = 1 | + (−1.22 − 0.707i)5-s + (1.22 − 0.707i)11-s + (−0.5 + 0.866i)13-s − 1.41i·17-s − 19-s + (−1.22 − 0.707i)23-s + (0.499 + 0.866i)25-s + (0.5 − 0.866i)31-s + (−0.5 − 0.866i)43-s + (0.5 − 0.866i)49-s + 1.41i·53-s − 2·55-s + (−1.22 − 0.707i)59-s + (−0.5 − 0.866i)61-s + (1.22 − 0.707i)65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1944 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.342 + 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1944 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.342 + 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6939975368\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6939975368\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (1.22 + 0.707i)T + (0.5 + 0.866i)T^{2} \) |
| 7 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 11 | \( 1 + (-1.22 + 0.707i)T + (0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 17 | \( 1 + 1.41iT - T^{2} \) |
| 19 | \( 1 + T + T^{2} \) |
| 23 | \( 1 + (1.22 + 0.707i)T + (0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + T^{2} \) |
| 41 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 43 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 47 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 53 | \( 1 - 1.41iT - T^{2} \) |
| 59 | \( 1 + (1.22 + 0.707i)T + (0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 + 1.41iT - T^{2} \) |
| 73 | \( 1 - T + T^{2} \) |
| 79 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + (1.22 - 0.707i)T + (0.5 - 0.866i)T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
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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
−9.053144897217665740406738937159, −8.399528076044576314963946034083, −7.68434296233102904451739541173, −6.79819241605179286032683173892, −6.08822179437168631041187103264, −4.76465979257542723738238454670, −4.28412435833198985062484393235, −3.47581811985778763780951573038, −2.08991354800683389986833557745, −0.50989876270657755568722498754,
1.66693811965743037216594596118, 3.00463704103505776743282196508, 3.94495546009474652113247351854, 4.38694697210790891595311974472, 5.77739898830749477299315078631, 6.59668990942052509348275524803, 7.27696510427596030188389703059, 8.066736865134777755175753312174, 8.596759389183322922424325466471, 9.782876699336677074660857741202