L(s) = 1 | + (0.139 − 1.40i)2-s + 0.319·3-s + (−1.96 − 0.391i)4-s + (1.20 − 1.88i)5-s + (0.0444 − 0.450i)6-s − i·7-s + (−0.823 + 2.70i)8-s − 2.89·9-s + (−2.48 − 1.95i)10-s − 4.31i·11-s + (−0.627 − 0.125i)12-s + 2.16·13-s + (−1.40 − 0.139i)14-s + (0.384 − 0.603i)15-s + (3.69 + 1.53i)16-s + 3.19i·17-s + ⋯ |
L(s) = 1 | + (0.0983 − 0.995i)2-s + 0.184·3-s + (−0.980 − 0.195i)4-s + (0.537 − 0.843i)5-s + (0.0181 − 0.183i)6-s − 0.377i·7-s + (−0.291 + 0.956i)8-s − 0.965·9-s + (−0.786 − 0.617i)10-s − 1.30i·11-s + (−0.181 − 0.0361i)12-s + 0.600·13-s + (−0.376 − 0.0371i)14-s + (0.0992 − 0.155i)15-s + (0.923 + 0.383i)16-s + 0.775i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.759 + 0.650i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.759 + 0.650i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.434352 - 1.17533i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.434352 - 1.17533i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.139 + 1.40i)T \) |
| 5 | \( 1 + (-1.20 + 1.88i)T \) |
| 7 | \( 1 + iT \) |
good | 3 | \( 1 - 0.319T + 3T^{2} \) |
| 11 | \( 1 + 4.31iT - 11T^{2} \) |
| 13 | \( 1 - 2.16T + 13T^{2} \) |
| 17 | \( 1 - 3.19iT - 17T^{2} \) |
| 19 | \( 1 + 5.38iT - 19T^{2} \) |
| 23 | \( 1 - 0.947iT - 23T^{2} \) |
| 29 | \( 1 - 7.55iT - 29T^{2} \) |
| 31 | \( 1 + 1.97T + 31T^{2} \) |
| 37 | \( 1 - 9.35T + 37T^{2} \) |
| 41 | \( 1 - 8.13T + 41T^{2} \) |
| 43 | \( 1 - 2.27T + 43T^{2} \) |
| 47 | \( 1 - 6.88iT - 47T^{2} \) |
| 53 | \( 1 - 6.12T + 53T^{2} \) |
| 59 | \( 1 + 4.30iT - 59T^{2} \) |
| 61 | \( 1 + 0.0705iT - 61T^{2} \) |
| 67 | \( 1 - 10.1T + 67T^{2} \) |
| 71 | \( 1 - 5.61T + 71T^{2} \) |
| 73 | \( 1 - 2.18iT - 73T^{2} \) |
| 79 | \( 1 - 10.0T + 79T^{2} \) |
| 83 | \( 1 + 16.4T + 83T^{2} \) |
| 89 | \( 1 + 13.4T + 89T^{2} \) |
| 97 | \( 1 - 11.4iT - 97T^{2} \) |
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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
−11.17106265461579610792848453703, −10.95622131521580435503196585805, −9.502365741075600056389924938441, −8.811974337347438150919110196073, −8.150423684554249152395827271633, −6.11342768080119087172888900151, −5.28251860640526799924424884198, −3.94478451381160323748289352826, −2.71499916710746273537119939890, −0.972859303285141930431100021733,
2.52511927699076133680665729823, 4.01980981582528602356318241885, 5.51648257102869936327641215793, 6.21424288261922467758629255716, 7.31246058971033329272447788465, 8.199482558455954097330274791903, 9.370418709207096016081847493336, 9.979175012600355149768379313630, 11.31339222642224326562614109987, 12.38076507365268564277254544082