| L(s) = 1 | + 2-s − 3-s + 4-s + 2·5-s − 6-s + 7-s + 8-s + 9-s + 2·10-s − 6·11-s − 12-s − 6·13-s + 14-s − 2·15-s + 16-s − 17-s + 18-s + 5·19-s + 2·20-s − 21-s − 6·22-s − 23-s − 24-s − 25-s − 6·26-s − 27-s + 28-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 − 1.80·11-s − 0.288·12-s − 1.66·13-s + 0.267·14-s − 0.516·15-s + 1/4·16-s − 0.242·17-s + 0.235·18-s + 1.14·19-s + 0.447·20-s − 0.218·21-s − 1.27·22-s − 0.208·23-s − 0.204·24-s − 1/5·25-s − 1.17·26-s − 0.192·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| 59 | \( 1 - T \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 11 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + 11 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + 5 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 15 T + p T^{2} \) |
| 89 | \( 1 - 15 T + p T^{2} \) |
| 97 | \( 1 + 17 T + p T^{2} \) |
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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
−7.74692117028269298062483067070, −6.91761065866668068663411696278, −6.04811730934370128530326738980, −5.47797791986377826720966031215, −4.91958229078526752396722629127, −4.46930114433166293223853855805, −2.94880292023662166139095650114, −2.52797988069657420362080098928, −1.53714322759938916014324390947, 0,
1.53714322759938916014324390947, 2.52797988069657420362080098928, 2.94880292023662166139095650114, 4.46930114433166293223853855805, 4.91958229078526752396722629127, 5.47797791986377826720966031215, 6.04811730934370128530326738980, 6.91761065866668068663411696278, 7.74692117028269298062483067070
