| L(s) = 1 | − 2-s + 3-s + 4-s + 5-s − 6-s − 8-s + 9-s − 10-s + 12-s − 13-s + 15-s + 16-s − 18-s + 20-s − 24-s + 25-s + 26-s + 27-s − 30-s − 32-s + 36-s − 39-s − 40-s + 2·41-s − 2·43-s + 45-s + 48-s + ⋯ |
| L(s) = 1 | − 2-s + 3-s + 4-s + 5-s − 6-s − 8-s + 9-s − 10-s + 12-s − 13-s + 15-s + 16-s − 18-s + 20-s − 24-s + 25-s + 26-s + 27-s − 30-s − 32-s + 36-s − 39-s − 40-s + 2·41-s − 2·43-s + 45-s + 48-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.198582535\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.198582535\) |
| \(L(1)\) |
|
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 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| good | 7 | \( 1 + T^{2} \) |
| 11 | \( ( 1 - T )( 1 + T ) \) |
| 17 | \( 1 + T^{2} \) |
| 19 | \( 1 + T^{2} \) |
| 23 | \( 1 + T^{2} \) |
| 29 | \( 1 + T^{2} \) |
| 31 | \( ( 1 - T )( 1 + T ) \) |
| 37 | \( ( 1 - T )( 1 + T ) \) |
| 41 | \( ( 1 - T )^{2} \) |
| 43 | \( ( 1 + T )^{2} \) |
| 47 | \( ( 1 - T )( 1 + T ) \) |
| 53 | \( ( 1 - T )( 1 + T ) \) |
| 59 | \( ( 1 - T )( 1 + T ) \) |
| 61 | \( ( 1 - T )( 1 + T ) \) |
| 67 | \( ( 1 - T )( 1 + T ) \) |
| 71 | \( ( 1 + T )^{2} \) |
| 73 | \( 1 + T^{2} \) |
| 79 | \( ( 1 + T )^{2} \) |
| 83 | \( ( 1 - T )^{2} \) |
| 89 | \( ( 1 + T )^{2} \) |
| 97 | \( 1 + T^{2} \) |
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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
−9.617491577200472130034388989993, −8.941424809775066615634410648681, −8.194792527080159563460027048944, −7.37310148028000106128810082902, −6.71614466108682143295487482295, −5.74993956813084240037296754846, −4.61456148074193800106382257507, −3.16313911805192908399185715721, −2.40660888618246790221497614286, −1.50135032922634249265141050625,
1.50135032922634249265141050625, 2.40660888618246790221497614286, 3.16313911805192908399185715721, 4.61456148074193800106382257507, 5.74993956813084240037296754846, 6.71614466108682143295487482295, 7.37310148028000106128810082902, 8.194792527080159563460027048944, 8.941424809775066615634410648681, 9.617491577200472130034388989993
