| L(s) = 1 | + 5·5-s + 28·7-s − 24·11-s − 70·13-s − 102·17-s − 20·19-s − 72·23-s + 25·25-s − 306·29-s + 136·31-s + 140·35-s − 214·37-s + 150·41-s + 292·43-s − 72·47-s + 441·49-s + 414·53-s − 120·55-s − 744·59-s − 418·61-s − 350·65-s − 188·67-s + 480·71-s + 434·73-s − 672·77-s − 1.35e3·79-s − 612·83-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 1.51·7-s − 0.657·11-s − 1.49·13-s − 1.45·17-s − 0.241·19-s − 0.652·23-s + 1/5·25-s − 1.95·29-s + 0.787·31-s + 0.676·35-s − 0.950·37-s + 0.571·41-s + 1.03·43-s − 0.223·47-s + 9/7·49-s + 1.07·53-s − 0.294·55-s − 1.64·59-s − 0.877·61-s − 0.667·65-s − 0.342·67-s + 0.802·71-s + 0.695·73-s − 0.994·77-s − 1.92·79-s − 0.809·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 - p T \) |
| good | 7 | \( 1 - 4 p T + p^{3} T^{2} \) |
| 11 | \( 1 + 24 T + p^{3} T^{2} \) |
| 13 | \( 1 + 70 T + p^{3} T^{2} \) |
| 17 | \( 1 + 6 p T + p^{3} T^{2} \) |
| 19 | \( 1 + 20 T + p^{3} T^{2} \) |
| 23 | \( 1 + 72 T + p^{3} T^{2} \) |
| 29 | \( 1 + 306 T + p^{3} T^{2} \) |
| 31 | \( 1 - 136 T + p^{3} T^{2} \) |
| 37 | \( 1 + 214 T + p^{3} T^{2} \) |
| 41 | \( 1 - 150 T + p^{3} T^{2} \) |
| 43 | \( 1 - 292 T + p^{3} T^{2} \) |
| 47 | \( 1 + 72 T + p^{3} T^{2} \) |
| 53 | \( 1 - 414 T + p^{3} T^{2} \) |
| 59 | \( 1 + 744 T + p^{3} T^{2} \) |
| 61 | \( 1 + 418 T + p^{3} T^{2} \) |
| 67 | \( 1 + 188 T + p^{3} T^{2} \) |
| 71 | \( 1 - 480 T + p^{3} T^{2} \) |
| 73 | \( 1 - 434 T + p^{3} T^{2} \) |
| 79 | \( 1 + 1352 T + p^{3} T^{2} \) |
| 83 | \( 1 + 612 T + p^{3} T^{2} \) |
| 89 | \( 1 - 30 T + p^{3} T^{2} \) |
| 97 | \( 1 + 286 T + p^{3} T^{2} \) |
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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
−9.568635028474010394600370619287, −8.716770483921667370122856666651, −7.79155753542801779077323565151, −7.15541737436847758901104467445, −5.84137283806300710110929303645, −4.97100939919980424854376217761, −4.28953158796361739287032141932, −2.49737499628677612533184955654, −1.80620269304406561305488064499, 0,
1.80620269304406561305488064499, 2.49737499628677612533184955654, 4.28953158796361739287032141932, 4.97100939919980424854376217761, 5.84137283806300710110929303645, 7.15541737436847758901104467445, 7.79155753542801779077323565151, 8.716770483921667370122856666651, 9.568635028474010394600370619287
