| L(s) = 1 | + 5-s + 5.29·11-s − 5.29·13-s − 3·17-s − 5.29·19-s − 4·25-s + 5.29·29-s + 5.29·31-s − 3·37-s − 9·41-s + 43-s + 47-s − 10.5·53-s + 5.29·55-s − 11·59-s + 5.29·61-s − 5.29·65-s + 4·67-s − 10.5·71-s − 10.5·73-s − 11·79-s − 9·83-s − 3·85-s + 14·89-s − 5.29·95-s + 5.29·97-s − 14·101-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 1.59·11-s − 1.46·13-s − 0.727·17-s − 1.21·19-s − 0.800·25-s + 0.982·29-s + 0.950·31-s − 0.493·37-s − 1.40·41-s + 0.152·43-s + 0.145·47-s − 1.45·53-s + 0.713·55-s − 1.43·59-s + 0.677·61-s − 0.656·65-s + 0.488·67-s − 1.25·71-s − 1.23·73-s − 1.23·79-s − 0.987·83-s − 0.325·85-s + 1.48·89-s − 0.542·95-s + 0.537·97-s − 1.39·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5292 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5292 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| good | 5 | \( 1 - T + 5T^{2} \) |
| 11 | \( 1 - 5.29T + 11T^{2} \) |
| 13 | \( 1 + 5.29T + 13T^{2} \) |
| 17 | \( 1 + 3T + 17T^{2} \) |
| 19 | \( 1 + 5.29T + 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 - 5.29T + 29T^{2} \) |
| 31 | \( 1 - 5.29T + 31T^{2} \) |
| 37 | \( 1 + 3T + 37T^{2} \) |
| 41 | \( 1 + 9T + 41T^{2} \) |
| 43 | \( 1 - T + 43T^{2} \) |
| 47 | \( 1 - T + 47T^{2} \) |
| 53 | \( 1 + 10.5T + 53T^{2} \) |
| 59 | \( 1 + 11T + 59T^{2} \) |
| 61 | \( 1 - 5.29T + 61T^{2} \) |
| 67 | \( 1 - 4T + 67T^{2} \) |
| 71 | \( 1 + 10.5T + 71T^{2} \) |
| 73 | \( 1 + 10.5T + 73T^{2} \) |
| 79 | \( 1 + 11T + 79T^{2} \) |
| 83 | \( 1 + 9T + 83T^{2} \) |
| 89 | \( 1 - 14T + 89T^{2} \) |
| 97 | \( 1 - 5.29T + 97T^{2} \) |
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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
−7.85338219160426333584214664868, −6.88444062675015463548541274714, −6.53479350408068373022768195150, −5.79983746799261645260968539151, −4.65410844528066104825460646584, −4.36915055241344551397727584495, −3.23070593469685790352629786395, −2.26637605135891887841922315757, −1.50544383537666756425540259959, 0,
1.50544383537666756425540259959, 2.26637605135891887841922315757, 3.23070593469685790352629786395, 4.36915055241344551397727584495, 4.65410844528066104825460646584, 5.79983746799261645260968539151, 6.53479350408068373022768195150, 6.88444062675015463548541274714, 7.85338219160426333584214664868
