| L(s) = 1 | + 5-s − 11-s + 4·13-s + 17-s + 5·19-s + 9·23-s + 25-s − 9·29-s + 2·31-s − 8·37-s − 2·41-s + 11·43-s − 2·47-s + 3·53-s − 55-s + 59-s − 11·61-s + 4·65-s − 6·67-s − 10·71-s + 12·73-s + 8·79-s − 9·83-s + 85-s + 9·89-s + 5·95-s − 97-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 0.301·11-s + 1.10·13-s + 0.242·17-s + 1.14·19-s + 1.87·23-s + 1/5·25-s − 1.67·29-s + 0.359·31-s − 1.31·37-s − 0.312·41-s + 1.67·43-s − 0.291·47-s + 0.412·53-s − 0.134·55-s + 0.130·59-s − 1.40·61-s + 0.496·65-s − 0.733·67-s − 1.18·71-s + 1.40·73-s + 0.900·79-s − 0.987·83-s + 0.108·85-s + 0.953·89-s + 0.512·95-s − 0.101·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 388080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 388080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.519360251\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.519360251\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 - 9 T + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 11 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 - T + p T^{2} \) |
| 61 | \( 1 + 11 T + p T^{2} \) |
| 67 | \( 1 + 6 T + p T^{2} \) |
| 71 | \( 1 + 10 T + p T^{2} \) |
| 73 | \( 1 - 12 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 9 T + p T^{2} \) |
| 89 | \( 1 - 9 T + p T^{2} \) |
| 97 | \( 1 + T + p T^{2} \) |
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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
−12.60158166076016, −12.01520950878690, −11.43857149588300, −11.09542599336001, −10.68124868047822, −10.33354567324990, −9.601897750273208, −9.321095774978115, −8.874382859796478, −8.513640754509735, −7.805672914844410, −7.355549932194067, −7.085551459489247, −6.385303521225234, −5.946991829143943, −5.408068980886421, −5.175393768295362, −4.517232007418774, −3.862453192860079, −3.306848186825800, −3.019658083339920, −2.304080313997285, −1.574225869155745, −1.188256972902572, −0.5062461897721424,
0.5062461897721424, 1.188256972902572, 1.574225869155745, 2.304080313997285, 3.019658083339920, 3.306848186825800, 3.862453192860079, 4.517232007418774, 5.175393768295362, 5.408068980886421, 5.946991829143943, 6.385303521225234, 7.085551459489247, 7.355549932194067, 7.805672914844410, 8.513640754509735, 8.874382859796478, 9.321095774978115, 9.601897750273208, 10.33354567324990, 10.68124868047822, 11.09542599336001, 11.43857149588300, 12.01520950878690, 12.60158166076016
