| L(s) = 1 | + 2-s + 4-s + 5-s + 7-s + 8-s + 10-s + 11-s − 4·13-s + 14-s + 16-s + 17-s − 2·19-s + 20-s + 22-s − 3·23-s + 25-s − 4·26-s + 28-s + 7·29-s − 5·31-s + 32-s + 34-s + 35-s − 2·38-s + 40-s − 10·41-s − 3·43-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.447·5-s + 0.377·7-s + 0.353·8-s + 0.316·10-s + 0.301·11-s − 1.10·13-s + 0.267·14-s + 1/4·16-s + 0.242·17-s − 0.458·19-s + 0.223·20-s + 0.213·22-s − 0.625·23-s + 1/5·25-s − 0.784·26-s + 0.188·28-s + 1.29·29-s − 0.898·31-s + 0.176·32-s + 0.171·34-s + 0.169·35-s − 0.324·38-s + 0.158·40-s − 1.56·41-s − 0.457·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16830 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16830 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| good | 7 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 - 7 T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 3 T + p T^{2} \) |
| 47 | \( 1 + 10 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + 14 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 16 T + p T^{2} \) |
| 71 | \( 1 + 16 T + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 + 2 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 - 13 T + p T^{2} \) |
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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
−16.17798325581398, −15.47245736077441, −14.93180818611630, −14.38780837792548, −14.14343903439421, −13.44612065377139, −12.80453776191997, −12.39738834867682, −11.74817589140686, −11.34850845451123, −10.54776050688113, −9.970239357575673, −9.633961380546186, −8.621693365570132, −8.211043091360962, −7.400053167596117, −6.821189667967901, −6.261474094483890, −5.556316302970144, −4.857010620688405, −4.523805091278334, −3.539362570448846, −2.897312877789631, −2.033544042274931, −1.457069240654718, 0,
1.457069240654718, 2.033544042274931, 2.897312877789631, 3.539362570448846, 4.523805091278334, 4.857010620688405, 5.556316302970144, 6.261474094483890, 6.821189667967901, 7.400053167596117, 8.211043091360962, 8.621693365570132, 9.633961380546186, 9.970239357575673, 10.54776050688113, 11.34850845451123, 11.74817589140686, 12.39738834867682, 12.80453776191997, 13.44612065377139, 14.14343903439421, 14.38780837792548, 14.93180818611630, 15.47245736077441, 16.17798325581398
