| L(s) = 1 | + 2-s + 4-s + 7-s + 8-s + 2·13-s + 14-s + 16-s − 2·19-s + 2·26-s + 28-s + 6·29-s + 8·31-s + 32-s + 4·37-s − 2·38-s + 6·41-s + 2·43-s − 6·47-s + 49-s + 2·52-s + 6·53-s + 56-s + 6·58-s − 12·59-s − 8·61-s + 8·62-s + 64-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.377·7-s + 0.353·8-s + 0.554·13-s + 0.267·14-s + 1/4·16-s − 0.458·19-s + 0.392·26-s + 0.188·28-s + 1.11·29-s + 1.43·31-s + 0.176·32-s + 0.657·37-s − 0.324·38-s + 0.937·41-s + 0.304·43-s − 0.875·47-s + 1/7·49-s + 0.277·52-s + 0.824·53-s + 0.133·56-s + 0.787·58-s − 1.56·59-s − 1.02·61-s + 1.01·62-s + 1/8·64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 381150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 381150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.963109180\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.963109180\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−12.51669187512400, −12.03162498372367, −11.63487599329556, −11.21856989737565, −10.71943169243180, −10.34143192647608, −9.922598743779086, −9.233964026972721, −8.825188277608064, −8.320049685197461, −7.779632578176238, −7.566204016870379, −6.748002143447976, −6.350786763361741, −6.070853107812410, −5.500429233347620, −4.750850177995040, −4.592813025902846, −4.088075438943511, −3.408778433508228, −2.922034874904114, −2.428950554316753, −1.793935745311200, −1.143616305321286, −0.5862774664770044,
0.5862774664770044, 1.143616305321286, 1.793935745311200, 2.428950554316753, 2.922034874904114, 3.408778433508228, 4.088075438943511, 4.592813025902846, 4.750850177995040, 5.500429233347620, 6.070853107812410, 6.350786763361741, 6.748002143447976, 7.566204016870379, 7.779632578176238, 8.320049685197461, 8.825188277608064, 9.233964026972721, 9.922598743779086, 10.34143192647608, 10.71943169243180, 11.21856989737565, 11.63487599329556, 12.03162498372367, 12.51669187512400
