L(s) = 1 | + 2-s + 3-s + 4-s + 6-s − 7-s + 8-s + 9-s + 12-s + 2·13-s − 14-s + 16-s + 6·17-s + 18-s − 21-s + 24-s + 2·26-s + 27-s − 28-s + 6·29-s + 4·31-s + 32-s + 6·34-s + 36-s + 2·37-s + 2·39-s − 6·41-s − 42-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s − 0.377·7-s + 0.353·8-s + 1/3·9-s + 0.288·12-s + 0.554·13-s − 0.267·14-s + 1/4·16-s + 1.45·17-s + 0.235·18-s − 0.218·21-s + 0.204·24-s + 0.392·26-s + 0.192·27-s − 0.188·28-s + 1.11·29-s + 0.718·31-s + 0.176·32-s + 1.02·34-s + 1/6·36-s + 0.328·37-s + 0.320·39-s − 0.937·41-s − 0.154·42-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 379050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 379050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(8.096091091\) |
\(L(\frac12)\) |
\(\approx\) |
\(8.096091091\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 19 | \( 1 \) |
good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−12.52858424328336, −11.96781676534381, −11.85543770082755, −11.24947330301362, −10.48445571021047, −10.31825841887664, −9.847461893702166, −9.403062296922275, −8.703322088239734, −8.276325580827215, −8.074797110591849, −7.234311061435757, −7.019373740486817, −6.466565607400721, −5.915855331681387, −5.480969106147009, −5.024303981918589, −4.326159595247818, −3.949406796468524, −3.357444329947455, −3.035021729375921, −2.475335495563117, −1.854715488516039, −1.113502878034518, −0.6794855803317969,
0.6794855803317969, 1.113502878034518, 1.854715488516039, 2.475335495563117, 3.035021729375921, 3.357444329947455, 3.949406796468524, 4.326159595247818, 5.024303981918589, 5.480969106147009, 5.915855331681387, 6.466565607400721, 7.019373740486817, 7.234311061435757, 8.074797110591849, 8.276325580827215, 8.703322088239734, 9.403062296922275, 9.847461893702166, 10.31825841887664, 10.48445571021047, 11.24947330301362, 11.85543770082755, 11.96781676534381, 12.52858424328336