| L(s) = 1 | − 2-s + 3-s + 5-s − 6-s + 7-s + 8-s − 10-s + 11-s − 2·13-s − 14-s + 15-s − 16-s + 17-s + 21-s − 22-s + 2·23-s + 24-s + 2·26-s − 27-s − 29-s − 30-s + 33-s − 34-s + 35-s − 2·39-s + 40-s + 41-s + ⋯ |
| L(s) = 1 | − 2-s + 3-s + 5-s − 6-s + 7-s + 8-s − 10-s + 11-s − 2·13-s − 14-s + 15-s − 16-s + 17-s + 21-s − 22-s + 2·23-s + 24-s + 2·26-s − 27-s − 29-s − 30-s + 33-s − 34-s + 35-s − 2·39-s + 40-s + 41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3311 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3311 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.302200188\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.302200188\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 43 | \( 1 - T \) |
| good | 2 | \( 1 + T + T^{2} \) |
| 3 | \( 1 - T + T^{2} \) |
| 5 | \( 1 - T + T^{2} \) |
| 13 | \( ( 1 + T )^{2} \) |
| 17 | \( 1 - T + T^{2} \) |
| 19 | \( ( 1 - T )( 1 + T ) \) |
| 23 | \( ( 1 - T )^{2} \) |
| 29 | \( 1 + T + T^{2} \) |
| 31 | \( ( 1 - T )( 1 + T ) \) |
| 37 | \( ( 1 - T )( 1 + T ) \) |
| 41 | \( 1 - T + T^{2} \) |
| 47 | \( ( 1 - T )( 1 + T ) \) |
| 53 | \( 1 + T + T^{2} \) |
| 59 | \( ( 1 - T )( 1 + T ) \) |
| 61 | \( ( 1 - T )( 1 + T ) \) |
| 67 | \( 1 + T + T^{2} \) |
| 71 | \( ( 1 - T )( 1 + T ) \) |
| 73 | \( ( 1 - T )( 1 + T ) \) |
| 79 | \( ( 1 - T )( 1 + T ) \) |
| 83 | \( 1 - T + T^{2} \) |
| 89 | \( ( 1 + T )^{2} \) |
| 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
−9.078329670282787940804988399391, −8.163160031254132457441558735612, −7.53082407583445598406375793862, −7.07945358053501462353042019144, −5.69071668128189538281756608925, −5.02243151355633050375769239669, −4.17726669530863524776272048742, −2.90066860169746725864157710471, −2.07659545032843478758638279190, −1.26249460765272589727504198734,
1.26249460765272589727504198734, 2.07659545032843478758638279190, 2.90066860169746725864157710471, 4.17726669530863524776272048742, 5.02243151355633050375769239669, 5.69071668128189538281756608925, 7.07945358053501462353042019144, 7.53082407583445598406375793862, 8.163160031254132457441558735612, 9.078329670282787940804988399391
