L(s) = 1 | + 2-s + 3-s + 4-s + 2·5-s + 6-s + 7-s + 8-s + 9-s + 2·10-s − 3·11-s + 12-s + 14-s + 2·15-s + 16-s − 7·17-s + 18-s + 5·19-s + 2·20-s + 21-s − 3·22-s + 6·23-s + 24-s − 25-s + 27-s + 28-s + 5·29-s + 2·30-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.894·5-s + 0.408·6-s + 0.377·7-s + 0.353·8-s + 1/3·9-s + 0.632·10-s − 0.904·11-s + 0.288·12-s + 0.267·14-s + 0.516·15-s + 1/4·16-s − 1.69·17-s + 0.235·18-s + 1.14·19-s + 0.447·20-s + 0.218·21-s − 0.639·22-s + 1.25·23-s + 0.204·24-s − 1/5·25-s + 0.192·27-s + 0.188·28-s + 0.928·29-s + 0.365·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7098 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7098 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(5.144791319\) |
\(L(\frac12)\) |
\(\approx\) |
\(5.144791319\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 \) |
good | 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 + 7 T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 5 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 5 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 - T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 7 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 12 T + p T^{2} \) |
| 79 | \( 1 - 3 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 - 11 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−7.75406846988640772463037995423, −7.25539193237494108999972531377, −6.44913590855609161752073485945, −5.77686360753249391609240974216, −4.95780173946931867166148360125, −4.55869765097645995538964666834, −3.48096397279521181697109399101, −2.57966516329149546025562647847, −2.21651120202354626578294616336, −1.04194818902661646583509526730,
1.04194818902661646583509526730, 2.21651120202354626578294616336, 2.57966516329149546025562647847, 3.48096397279521181697109399101, 4.55869765097645995538964666834, 4.95780173946931867166148360125, 5.77686360753249391609240974216, 6.44913590855609161752073485945, 7.25539193237494108999972531377, 7.75406846988640772463037995423