| L(s) = 1 | + 2-s + 3-s + 4-s − 3·5-s + 6-s + 7-s + 8-s + 9-s − 3·10-s + 3·11-s + 12-s + 5·13-s + 14-s − 3·15-s + 16-s − 17-s + 18-s + 2·19-s − 3·20-s + 21-s + 3·22-s + 6·23-s + 24-s + 4·25-s + 5·26-s + 27-s + 28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s − 1.34·5-s + 0.408·6-s + 0.377·7-s + 0.353·8-s + 1/3·9-s − 0.948·10-s + 0.904·11-s + 0.288·12-s + 1.38·13-s + 0.267·14-s − 0.774·15-s + 1/4·16-s − 0.242·17-s + 0.235·18-s + 0.458·19-s − 0.670·20-s + 0.218·21-s + 0.639·22-s + 1.25·23-s + 0.204·24-s + 4/5·25-s + 0.980·26-s + 0.192·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 714 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 714 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.592991985\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.592991985\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| good | 5 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 11 T + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 7 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 + 15 T + p T^{2} \) |
| 89 | \( 1 - 9 T + p T^{2} \) |
| 97 | \( 1 + 19 T + p T^{2} \) |
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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
−10.87935945692914470065617822189, −9.343140110042709561392981557520, −8.613071818417961118960922066933, −7.73220941821340135084187432759, −7.03852758920525243298654053793, −5.95409201205149097714963015953, −4.62735512824122009930531903861, −3.85240346236466809058448583980, −3.16932259931988009518519746082, −1.41242540834598329616667848501,
1.41242540834598329616667848501, 3.16932259931988009518519746082, 3.85240346236466809058448583980, 4.62735512824122009930531903861, 5.95409201205149097714963015953, 7.03852758920525243298654053793, 7.73220941821340135084187432759, 8.613071818417961118960922066933, 9.343140110042709561392981557520, 10.87935945692914470065617822189
