| L(s) = 1 | + 5-s − 4·7-s + 6·11-s + 4·13-s − 4·17-s + 8·19-s + 25-s + 2·29-s − 2·31-s − 4·35-s − 4·37-s − 6·41-s + 12·43-s + 9·49-s − 14·53-s + 6·55-s − 6·59-s + 6·61-s + 4·65-s + 4·67-s + 8·71-s − 2·73-s − 24·77-s + 6·79-s + 12·83-s − 4·85-s − 6·89-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 1.51·7-s + 1.80·11-s + 1.10·13-s − 0.970·17-s + 1.83·19-s + 1/5·25-s + 0.371·29-s − 0.359·31-s − 0.676·35-s − 0.657·37-s − 0.937·41-s + 1.82·43-s + 9/7·49-s − 1.92·53-s + 0.809·55-s − 0.781·59-s + 0.768·61-s + 0.496·65-s + 0.488·67-s + 0.949·71-s − 0.234·73-s − 2.73·77-s + 0.675·79-s + 1.31·83-s − 0.433·85-s − 0.635·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.171802405\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.171802405\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 14 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 6 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 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
−8.233707745846371238456044860170, −7.09752660423814295441357875001, −6.63051434487844039149409502075, −6.16160510715634515135325957529, −5.42672318398502146703592000043, −4.28389757470006853731361576226, −3.55192887194675292853371851200, −3.04129716998461074149844302940, −1.74072122715861936107235590781, −0.818171558753717111286502894239,
0.818171558753717111286502894239, 1.74072122715861936107235590781, 3.04129716998461074149844302940, 3.55192887194675292853371851200, 4.28389757470006853731361576226, 5.42672318398502146703592000043, 6.16160510715634515135325957529, 6.63051434487844039149409502075, 7.09752660423814295441357875001, 8.233707745846371238456044860170
