| L(s) = 1 | − 5-s − 7-s − 3·13-s + 2·17-s + 5·19-s + 8·23-s − 4·25-s + 7·29-s − 8·31-s + 35-s − 3·37-s − 10·41-s + 10·43-s − 7·47-s + 49-s + 2·53-s + 9·59-s + 2·61-s + 3·65-s − 3·67-s − 6·71-s − 73-s − 10·79-s − 6·83-s − 2·85-s − 2·89-s + 3·91-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.377·7-s − 0.832·13-s + 0.485·17-s + 1.14·19-s + 1.66·23-s − 4/5·25-s + 1.29·29-s − 1.43·31-s + 0.169·35-s − 0.493·37-s − 1.56·41-s + 1.52·43-s − 1.02·47-s + 1/7·49-s + 0.274·53-s + 1.17·59-s + 0.256·61-s + 0.372·65-s − 0.366·67-s − 0.712·71-s − 0.117·73-s − 1.12·79-s − 0.658·83-s − 0.216·85-s − 0.211·89-s + 0.314·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 60984 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60984 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 5 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 7 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 + 7 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 9 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 3 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−14.46392486121947, −14.16634982403406, −13.50388927321314, −12.97379025161474, −12.53896573932557, −11.99248706802022, −11.55005219777013, −11.13717630786397, −10.30040858579209, −10.03398643274454, −9.444827343443013, −8.887299502719509, −8.411294580811502, −7.648481275508674, −7.223745976602218, −6.933590272418969, −6.103277470696864, −5.398537796534604, −5.076953520694589, −4.375996021173067, −3.614104653795494, −3.147197486025564, −2.588588164913750, −1.645200312231988, −0.8791320798993977, 0,
0.8791320798993977, 1.645200312231988, 2.588588164913750, 3.147197486025564, 3.614104653795494, 4.375996021173067, 5.076953520694589, 5.398537796534604, 6.103277470696864, 6.933590272418969, 7.223745976602218, 7.648481275508674, 8.411294580811502, 8.887299502719509, 9.444827343443013, 10.03398643274454, 10.30040858579209, 11.13717630786397, 11.55005219777013, 11.99248706802022, 12.53896573932557, 12.97379025161474, 13.50388927321314, 14.16634982403406, 14.46392486121947
