| L(s) = 1 | − 2·2-s − 3·3-s + 4·4-s + 5·5-s + 6·6-s + 16·7-s − 8·8-s + 9·9-s − 10·10-s + 4·11-s − 12·12-s − 26·13-s − 32·14-s − 15·15-s + 16·16-s − 30·17-s − 18·18-s − 100·19-s + 20·20-s − 48·21-s − 8·22-s − 23·23-s + 24·24-s + 25·25-s + 52·26-s − 27·27-s + 64·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s + 0.863·7-s − 0.353·8-s + 1/3·9-s − 0.316·10-s + 0.109·11-s − 0.288·12-s − 0.554·13-s − 0.610·14-s − 0.258·15-s + 1/4·16-s − 0.428·17-s − 0.235·18-s − 1.20·19-s + 0.223·20-s − 0.498·21-s − 0.0775·22-s − 0.208·23-s + 0.204·24-s + 1/5·25-s + 0.392·26-s − 0.192·27-s + 0.431·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p T \) |
| 3 | \( 1 + p T \) |
| 5 | \( 1 - p T \) |
| 23 | \( 1 + p T \) |
| good | 7 | \( 1 - 16 T + p^{3} T^{2} \) |
| 11 | \( 1 - 4 T + p^{3} T^{2} \) |
| 13 | \( 1 + 2 p T + p^{3} T^{2} \) |
| 17 | \( 1 + 30 T + p^{3} T^{2} \) |
| 19 | \( 1 + 100 T + p^{3} T^{2} \) |
| 29 | \( 1 - 94 T + p^{3} T^{2} \) |
| 31 | \( 1 + 232 T + p^{3} T^{2} \) |
| 37 | \( 1 - 230 T + p^{3} T^{2} \) |
| 41 | \( 1 + 150 T + p^{3} T^{2} \) |
| 43 | \( 1 + 156 T + p^{3} T^{2} \) |
| 47 | \( 1 - 544 T + p^{3} T^{2} \) |
| 53 | \( 1 + 34 T + p^{3} T^{2} \) |
| 59 | \( 1 + 388 T + p^{3} T^{2} \) |
| 61 | \( 1 - 174 T + p^{3} T^{2} \) |
| 67 | \( 1 + 484 T + p^{3} T^{2} \) |
| 71 | \( 1 - 440 T + p^{3} T^{2} \) |
| 73 | \( 1 + 550 T + p^{3} T^{2} \) |
| 79 | \( 1 - 376 T + p^{3} T^{2} \) |
| 83 | \( 1 - 652 T + p^{3} T^{2} \) |
| 89 | \( 1 + 1350 T + p^{3} T^{2} \) |
| 97 | \( 1 + 542 T + p^{3} T^{2} \) |
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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
−9.667419062241901770306373109435, −8.820700839338852450755283282014, −7.959523048645634439800788052476, −7.01998334178709121509938054029, −6.16456491221778499836401836427, −5.18337485588622598164344478893, −4.19104812111194959318743810500, −2.44265122640732001259296299414, −1.45900197536789464853170292589, 0,
1.45900197536789464853170292589, 2.44265122640732001259296299414, 4.19104812111194959318743810500, 5.18337485588622598164344478893, 6.16456491221778499836401836427, 7.01998334178709121509938054029, 7.959523048645634439800788052476, 8.820700839338852450755283282014, 9.667419062241901770306373109435
