| L(s) = 1 | − 4·7-s + 2·11-s − 6·13-s − 2·17-s − 19-s − 6·23-s − 8·29-s + 8·31-s − 10·37-s + 4·41-s + 4·43-s − 6·47-s + 9·49-s + 12·53-s − 8·59-s + 2·61-s + 4·67-s + 12·71-s + 10·73-s − 8·77-s + 8·79-s − 2·83-s + 12·89-s + 24·91-s + 14·97-s + 101-s + 103-s + ⋯ |
| L(s) = 1 | − 1.51·7-s + 0.603·11-s − 1.66·13-s − 0.485·17-s − 0.229·19-s − 1.25·23-s − 1.48·29-s + 1.43·31-s − 1.64·37-s + 0.624·41-s + 0.609·43-s − 0.875·47-s + 9/7·49-s + 1.64·53-s − 1.04·59-s + 0.256·61-s + 0.488·67-s + 1.42·71-s + 1.17·73-s − 0.911·77-s + 0.900·79-s − 0.219·83-s + 1.27·89-s + 2.51·91-s + 1.42·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 68400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 68400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 4 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 2 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−14.36553725489250, −13.89312677979031, −13.42018279765648, −12.86088593786913, −12.27427675779346, −12.16475377247884, −11.52928441323280, −10.79986723139690, −10.18217143491335, −9.898703151306828, −9.285185901573059, −9.108603662790901, −8.207269491745188, −7.687352728252688, −7.067592933959059, −6.582619187653926, −6.259860874605243, −5.476858239712131, −4.978178065429219, −4.120593115377330, −3.782638105159374, −3.059771332235521, −2.357924156791244, −1.927365343718279, −0.6600856253612702, 0,
0.6600856253612702, 1.927365343718279, 2.357924156791244, 3.059771332235521, 3.782638105159374, 4.120593115377330, 4.978178065429219, 5.476858239712131, 6.259860874605243, 6.582619187653926, 7.067592933959059, 7.687352728252688, 8.207269491745188, 9.108603662790901, 9.285185901573059, 9.898703151306828, 10.18217143491335, 10.79986723139690, 11.52928441323280, 12.16475377247884, 12.27427675779346, 12.86088593786913, 13.42018279765648, 13.89312677979031, 14.36553725489250
