| L(s) = 1 | + 3·5-s + 7-s + 6·11-s + 13-s + 3·17-s − 2·19-s + 4·25-s − 6·29-s + 4·31-s + 3·35-s − 7·37-s + 43-s + 3·47-s − 6·49-s + 18·55-s − 6·59-s + 8·61-s + 3·65-s − 14·67-s − 3·71-s + 2·73-s + 6·77-s − 8·79-s + 12·83-s + 9·85-s + 6·89-s + 91-s + ⋯ |
| L(s) = 1 | + 1.34·5-s + 0.377·7-s + 1.80·11-s + 0.277·13-s + 0.727·17-s − 0.458·19-s + 4/5·25-s − 1.11·29-s + 0.718·31-s + 0.507·35-s − 1.15·37-s + 0.152·43-s + 0.437·47-s − 6/7·49-s + 2.42·55-s − 0.781·59-s + 1.02·61-s + 0.372·65-s − 1.71·67-s − 0.356·71-s + 0.234·73-s + 0.683·77-s − 0.900·79-s + 1.31·83-s + 0.976·85-s + 0.635·89-s + 0.104·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.679014865\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.679014865\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 5 | \( 1 - 3 T + p T^{2} \) |
| 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + 14 T + p T^{2} \) |
| 71 | \( 1 + 3 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−9.197451332644930457875280868446, −8.727543231328509862646305089705, −7.62631082392611378394973504804, −6.63805072209796513425044648809, −6.10177969443293897115150871480, −5.33155182765537854352160831380, −4.28669590144044996642748248504, −3.33995220257029308986401899415, −1.98547611328597681321826282171, −1.28946814831643962415194675902,
1.28946814831643962415194675902, 1.98547611328597681321826282171, 3.33995220257029308986401899415, 4.28669590144044996642748248504, 5.33155182765537854352160831380, 6.10177969443293897115150871480, 6.63805072209796513425044648809, 7.62631082392611378394973504804, 8.727543231328509862646305089705, 9.197451332644930457875280868446
