| L(s) = 1 | − 2-s + 4-s − 5-s − 7-s − 8-s + 10-s − 11-s + 14-s + 16-s + 17-s − 19-s − 20-s + 22-s + 25-s − 28-s + 29-s − 31-s − 32-s − 34-s + 35-s − 37-s + 38-s + 40-s − 44-s − 50-s + 53-s + 55-s + ⋯ |
| L(s) = 1 | − 2-s + 4-s − 5-s − 7-s − 8-s + 10-s − 11-s + 14-s + 16-s + 17-s − 19-s − 20-s + 22-s + 25-s − 28-s + 29-s − 31-s − 32-s − 34-s + 35-s − 37-s + 38-s + 40-s − 44-s − 50-s + 53-s + 55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4342611629\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4342611629\) |
| \(L(1)\) |
|
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| good | 7 | \( 1 + T + T^{2} \) |
| 13 | \( ( 1 - T )( 1 + T ) \) |
| 17 | \( 1 - T + T^{2} \) |
| 19 | \( 1 + T + T^{2} \) |
| 23 | \( ( 1 - T )( 1 + T ) \) |
| 29 | \( 1 - T + T^{2} \) |
| 31 | \( 1 + T + T^{2} \) |
| 37 | \( 1 + T + T^{2} \) |
| 41 | \( ( 1 - T )( 1 + T ) \) |
| 43 | \( ( 1 - T )( 1 + T ) \) |
| 47 | \( ( 1 - T )( 1 + T ) \) |
| 53 | \( 1 - T + T^{2} \) |
| 59 | \( ( 1 - T )( 1 + T ) \) |
| 61 | \( 1 + T + T^{2} \) |
| 67 | \( ( 1 - T )^{2} \) |
| 71 | \( 1 - T + T^{2} \) |
| 73 | \( ( 1 - T )^{2} \) |
| 79 | \( ( 1 - T )( 1 + T ) \) |
| 83 | \( ( 1 - T )( 1 + T ) \) |
| 89 | \( 1 - T + T^{2} \) |
| 97 | \( ( 1 - T )( 1 + T ) \) |
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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
−8.403124144176362977837487646191, −8.130192596445280300039096351036, −7.22731585055239551086429339225, −6.75317205369445041414481855172, −5.86664005744627583874031154545, −4.98051285202963894221363603126, −3.70477362547386110863939095350, −3.14587942267611780900500058358, −2.17926926906185071217829750252, −0.61415038615309546718295561118,
0.61415038615309546718295561118, 2.17926926906185071217829750252, 3.14587942267611780900500058358, 3.70477362547386110863939095350, 4.98051285202963894221363603126, 5.86664005744627583874031154545, 6.75317205369445041414481855172, 7.22731585055239551086429339225, 8.130192596445280300039096351036, 8.403124144176362977837487646191
