| L(s) = 1 | − 2-s − 3-s − 4-s + 5-s + 6-s + 3·8-s + 9-s − 10-s + 12-s + 6·13-s − 15-s − 16-s + 17-s − 18-s − 4·19-s − 20-s − 3·24-s + 25-s − 6·26-s − 27-s − 2·29-s + 30-s − 5·32-s − 34-s − 36-s − 6·37-s + 4·38-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s − 1/2·4-s + 0.447·5-s + 0.408·6-s + 1.06·8-s + 1/3·9-s − 0.316·10-s + 0.288·12-s + 1.66·13-s − 0.258·15-s − 1/4·16-s + 0.242·17-s − 0.235·18-s − 0.917·19-s − 0.223·20-s − 0.612·24-s + 1/5·25-s − 1.17·26-s − 0.192·27-s − 0.371·29-s + 0.182·30-s − 0.883·32-s − 0.171·34-s − 1/6·36-s − 0.986·37-s + 0.648·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12495 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12495 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.126408931\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.126408931\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 2 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
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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
−16.48682835039136, −15.91006719949705, −15.42107096101159, −14.44361472329198, −14.10012014244937, −13.36614607785034, −12.93827600314384, −12.50760394667210, −11.42534772436336, −11.12481806551466, −10.35572882420666, −10.15251128525180, −9.196188974401518, −8.852969701856337, −8.263211006970189, −7.563104601721418, −6.821975894006663, −6.083198932826379, −5.634889701822202, −4.815807905717032, −4.087091892185414, −3.504680186699925, −2.201177524863629, −1.373421350231584, −0.6198493776630361,
0.6198493776630361, 1.373421350231584, 2.201177524863629, 3.504680186699925, 4.087091892185414, 4.815807905717032, 5.634889701822202, 6.083198932826379, 6.821975894006663, 7.563104601721418, 8.263211006970189, 8.852969701856337, 9.196188974401518, 10.15251128525180, 10.35572882420666, 11.12481806551466, 11.42534772436336, 12.50760394667210, 12.93827600314384, 13.36614607785034, 14.10012014244937, 14.44361472329198, 15.42107096101159, 15.91006719949705, 16.48682835039136
