| L(s) = 1 | + 2-s − 3-s + 4-s + 5-s − 6-s − 7-s + 8-s + 9-s + 10-s + 11-s − 12-s − 3·13-s − 14-s − 15-s + 16-s + 18-s + 8·19-s + 20-s + 21-s + 22-s + 4·23-s − 24-s + 25-s − 3·26-s − 27-s − 28-s − 2·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s − 0.377·7-s + 0.353·8-s + 1/3·9-s + 0.316·10-s + 0.301·11-s − 0.288·12-s − 0.832·13-s − 0.267·14-s − 0.258·15-s + 1/4·16-s + 0.235·18-s + 1.83·19-s + 0.223·20-s + 0.218·21-s + 0.213·22-s + 0.834·23-s − 0.204·24-s + 1/5·25-s − 0.588·26-s − 0.192·27-s − 0.188·28-s − 0.371·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 60690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 17 | \( 1 \) |
| good | 11 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 7 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 + T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 14 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 3 T + p T^{2} \) |
| 71 | \( 1 - 3 T + p T^{2} \) |
| 73 | \( 1 - 9 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 89 | \( 1 + 9 T + p T^{2} \) |
| 97 | \( 1 + T + p T^{2} \) |
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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
−14.62731955468013, −13.89726535179607, −13.49335985361445, −13.15232708925430, −12.40307361867715, −12.18035514387367, −11.54347545325689, −11.18261530914903, −10.55651578168611, −9.928419233258302, −9.525837552092806, −9.170933117123196, −8.199270085506618, −7.628715832548708, −6.950703085684533, −6.782365934990297, −6.004971248119804, −5.417113568768841, −5.098713346084468, −4.562607668329405, −3.649435552126614, −3.227858071329191, −2.551109292225243, −1.697415782843288, −1.076257807764240, 0,
1.076257807764240, 1.697415782843288, 2.551109292225243, 3.227858071329191, 3.649435552126614, 4.562607668329405, 5.098713346084468, 5.417113568768841, 6.004971248119804, 6.782365934990297, 6.950703085684533, 7.628715832548708, 8.199270085506618, 9.170933117123196, 9.525837552092806, 9.928419233258302, 10.55651578168611, 11.18261530914903, 11.54347545325689, 12.18035514387367, 12.40307361867715, 13.15232708925430, 13.49335985361445, 13.89726535179607, 14.62731955468013
