L(s) = 1 | − 2-s − 3-s + 4-s − 4.29·5-s + 6-s + 0.914·7-s − 8-s + 9-s + 4.29·10-s + 2.25·11-s − 12-s + 4.58·13-s − 0.914·14-s + 4.29·15-s + 16-s + 17-s − 18-s + 6.28·19-s − 4.29·20-s − 0.914·21-s − 2.25·22-s + 6.33·23-s + 24-s + 13.4·25-s − 4.58·26-s − 27-s + 0.914·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s − 1.92·5-s + 0.408·6-s + 0.345·7-s − 0.353·8-s + 0.333·9-s + 1.35·10-s + 0.679·11-s − 0.288·12-s + 1.27·13-s − 0.244·14-s + 1.10·15-s + 0.250·16-s + 0.242·17-s − 0.235·18-s + 1.44·19-s − 0.960·20-s − 0.199·21-s − 0.480·22-s + 1.32·23-s + 0.204·24-s + 2.69·25-s − 0.899·26-s − 0.192·27-s + 0.172·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.098787392\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.098787392\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| 59 | \( 1 + T \) |
good | 5 | \( 1 + 4.29T + 5T^{2} \) |
| 7 | \( 1 - 0.914T + 7T^{2} \) |
| 11 | \( 1 - 2.25T + 11T^{2} \) |
| 13 | \( 1 - 4.58T + 13T^{2} \) |
| 19 | \( 1 - 6.28T + 19T^{2} \) |
| 23 | \( 1 - 6.33T + 23T^{2} \) |
| 29 | \( 1 - 5.98T + 29T^{2} \) |
| 31 | \( 1 - 9.32T + 31T^{2} \) |
| 37 | \( 1 + 0.672T + 37T^{2} \) |
| 41 | \( 1 + 2.94T + 41T^{2} \) |
| 43 | \( 1 - 6.03T + 43T^{2} \) |
| 47 | \( 1 + 7.72T + 47T^{2} \) |
| 53 | \( 1 - 2.77T + 53T^{2} \) |
| 61 | \( 1 - 5.17T + 61T^{2} \) |
| 67 | \( 1 + 11.1T + 67T^{2} \) |
| 71 | \( 1 + 3.97T + 71T^{2} \) |
| 73 | \( 1 - 11.9T + 73T^{2} \) |
| 79 | \( 1 + 12.5T + 79T^{2} \) |
| 83 | \( 1 - 14.8T + 83T^{2} \) |
| 89 | \( 1 - 13.9T + 89T^{2} \) |
| 97 | \( 1 + 0.989T + 97T^{2} \) |
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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
−8.107881182880268070331719657300, −7.46874239945158381569466109429, −6.81533169906545855380725188861, −6.23003472991702419107185148793, −5.05874225221745987674190564802, −4.47603362344392666463835185242, −3.53255604925099414285945932314, −3.03595385511793322069551757334, −1.19973792873057650811596414198, −0.794875533242016920508969725230,
0.794875533242016920508969725230, 1.19973792873057650811596414198, 3.03595385511793322069551757334, 3.53255604925099414285945932314, 4.47603362344392666463835185242, 5.05874225221745987674190564802, 6.23003472991702419107185148793, 6.81533169906545855380725188861, 7.46874239945158381569466109429, 8.107881182880268070331719657300