| L(s) = 1 | + 5-s + 11-s − 13-s − 3·17-s + 3·19-s + 2·23-s + 25-s + 4·29-s − 6·31-s + 3·37-s + 7·41-s + 7·43-s − 9·47-s − 7·49-s + 55-s − 6·59-s + 6·61-s − 65-s − 7·67-s + 10·71-s + 4·73-s − 6·79-s − 2·83-s − 3·85-s − 18·89-s + 3·95-s − 7·97-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 0.301·11-s − 0.277·13-s − 0.727·17-s + 0.688·19-s + 0.417·23-s + 1/5·25-s + 0.742·29-s − 1.07·31-s + 0.493·37-s + 1.09·41-s + 1.06·43-s − 1.31·47-s − 49-s + 0.134·55-s − 0.781·59-s + 0.768·61-s − 0.124·65-s − 0.855·67-s + 1.18·71-s + 0.468·73-s − 0.675·79-s − 0.219·83-s − 0.325·85-s − 1.90·89-s + 0.307·95-s − 0.710·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 102960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 102960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - 3 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 - 7 T + p T^{2} \) |
| 43 | \( 1 - 7 T + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 7 T + p T^{2} \) |
| 71 | \( 1 - 10 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + 6 T + p T^{2} \) |
| 83 | \( 1 + 2 T + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 97 | \( 1 + 7 T + p T^{2} \) |
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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
−13.98984204403167, −13.53774814350098, −12.83639866600619, −12.69879393540938, −12.07071197817237, −11.37336919018250, −11.09564317833466, −10.64916588546157, −9.801503288059342, −9.629699184634711, −9.112682958539229, −8.545124993356244, −8.013482130610134, −7.383366816247990, −6.934754865569393, −6.400474303661365, −5.843327619413101, −5.329715578743645, −4.692480320735231, −4.258689653864494, −3.498069013786561, −2.882288113850483, −2.334490560010897, −1.591901429278130, −0.9594755234430114, 0,
0.9594755234430114, 1.591901429278130, 2.334490560010897, 2.882288113850483, 3.498069013786561, 4.258689653864494, 4.692480320735231, 5.329715578743645, 5.843327619413101, 6.400474303661365, 6.934754865569393, 7.383366816247990, 8.013482130610134, 8.545124993356244, 9.112682958539229, 9.629699184634711, 9.801503288059342, 10.64916588546157, 11.09564317833466, 11.37336919018250, 12.07071197817237, 12.69879393540938, 12.83639866600619, 13.53774814350098, 13.98984204403167
