| L(s) = 1 | + 25·5-s + 100·7-s − 136·11-s + 82·13-s − 358·17-s − 796·19-s + 488·23-s + 625·25-s − 7.46e3·29-s − 2.72e3·31-s + 2.50e3·35-s + 7.79e3·37-s − 1.82e4·41-s + 2.44e3·43-s − 2.20e3·47-s − 6.80e3·49-s − 1.01e4·53-s − 3.40e3·55-s − 6.77e3·59-s + 2.33e4·61-s + 2.05e3·65-s + 9.67e3·67-s + 1.37e4·71-s − 2.73e4·73-s − 1.36e4·77-s + 9.32e4·79-s − 2.32e4·83-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 0.771·7-s − 0.338·11-s + 0.134·13-s − 0.300·17-s − 0.505·19-s + 0.192·23-s + 1/5·25-s − 1.64·29-s − 0.509·31-s + 0.344·35-s + 0.935·37-s − 1.69·41-s + 0.201·43-s − 0.145·47-s − 0.405·49-s − 0.494·53-s − 0.151·55-s − 0.253·59-s + 0.805·61-s + 0.0601·65-s + 0.263·67-s + 0.323·71-s − 0.601·73-s − 0.261·77-s + 1.68·79-s − 0.370·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 - p^{2} T \) |
| good | 7 | \( 1 - 100 T + p^{5} T^{2} \) |
| 11 | \( 1 + 136 T + p^{5} T^{2} \) |
| 13 | \( 1 - 82 T + p^{5} T^{2} \) |
| 17 | \( 1 + 358 T + p^{5} T^{2} \) |
| 19 | \( 1 + 796 T + p^{5} T^{2} \) |
| 23 | \( 1 - 488 T + p^{5} T^{2} \) |
| 29 | \( 1 + 7466 T + p^{5} T^{2} \) |
| 31 | \( 1 + 88 p T + p^{5} T^{2} \) |
| 37 | \( 1 - 7794 T + p^{5} T^{2} \) |
| 41 | \( 1 + 18234 T + p^{5} T^{2} \) |
| 43 | \( 1 - 2444 T + p^{5} T^{2} \) |
| 47 | \( 1 + 2200 T + p^{5} T^{2} \) |
| 53 | \( 1 + 10122 T + p^{5} T^{2} \) |
| 59 | \( 1 + 6776 T + p^{5} T^{2} \) |
| 61 | \( 1 - 23398 T + p^{5} T^{2} \) |
| 67 | \( 1 - 9676 T + p^{5} T^{2} \) |
| 71 | \( 1 - 13728 T + p^{5} T^{2} \) |
| 73 | \( 1 + 27390 T + p^{5} T^{2} \) |
| 79 | \( 1 - 93288 T + p^{5} T^{2} \) |
| 83 | \( 1 + 23276 T + p^{5} T^{2} \) |
| 89 | \( 1 + 102354 T + p^{5} T^{2} \) |
| 97 | \( 1 + 49502 T + p^{5} T^{2} \) |
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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
−9.206457201176154651035245997244, −8.363930174271021889915784706452, −7.54794079077097008282990594513, −6.54968266415927979157092989184, −5.55558075446428836689411629317, −4.77292866137240781345924355425, −3.64854011328815478353481227954, −2.33176665072683094760727213543, −1.45800253882313486615557033272, 0,
1.45800253882313486615557033272, 2.33176665072683094760727213543, 3.64854011328815478353481227954, 4.77292866137240781345924355425, 5.55558075446428836689411629317, 6.54968266415927979157092989184, 7.54794079077097008282990594513, 8.363930174271021889915784706452, 9.206457201176154651035245997244
