| L(s) = 1 | + 5-s − 2·7-s − 13-s + 2.47·17-s + 2·19-s − 6.47·23-s + 25-s − 2.47·29-s − 4.47·31-s − 2·35-s + 10.9·37-s − 6.94·41-s − 4·47-s − 3·49-s + 8.94·53-s + 4.94·59-s − 10.9·61-s − 65-s − 12.4·67-s + 8.94·71-s − 12.4·73-s + 0.944·79-s − 8·83-s + 2.47·85-s + 2·89-s + 2·91-s + 2·95-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 0.755·7-s − 0.277·13-s + 0.599·17-s + 0.458·19-s − 1.34·23-s + 0.200·25-s − 0.459·29-s − 0.803·31-s − 0.338·35-s + 1.79·37-s − 1.08·41-s − 0.583·47-s − 0.428·49-s + 1.22·53-s + 0.643·59-s − 1.40·61-s − 0.124·65-s − 1.52·67-s + 1.06·71-s − 1.45·73-s + 0.106·79-s − 0.878·83-s + 0.268·85-s + 0.211·89-s + 0.209·91-s + 0.205·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| good | 7 | \( 1 + 2T + 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 17 | \( 1 - 2.47T + 17T^{2} \) |
| 19 | \( 1 - 2T + 19T^{2} \) |
| 23 | \( 1 + 6.47T + 23T^{2} \) |
| 29 | \( 1 + 2.47T + 29T^{2} \) |
| 31 | \( 1 + 4.47T + 31T^{2} \) |
| 37 | \( 1 - 10.9T + 37T^{2} \) |
| 41 | \( 1 + 6.94T + 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 + 4T + 47T^{2} \) |
| 53 | \( 1 - 8.94T + 53T^{2} \) |
| 59 | \( 1 - 4.94T + 59T^{2} \) |
| 61 | \( 1 + 10.9T + 61T^{2} \) |
| 67 | \( 1 + 12.4T + 67T^{2} \) |
| 71 | \( 1 - 8.94T + 71T^{2} \) |
| 73 | \( 1 + 12.4T + 73T^{2} \) |
| 79 | \( 1 - 0.944T + 79T^{2} \) |
| 83 | \( 1 + 8T + 83T^{2} \) |
| 89 | \( 1 - 2T + 89T^{2} \) |
| 97 | \( 1 + 4.47T + 97T^{2} \) |
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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
−7.83186370683019467591166338181, −7.29004645934603250674463684080, −6.34518743211658350505908566092, −5.86640160768240914318904531858, −5.09062225308872914362814185000, −4.09884680947828630682044141523, −3.31040685371487920512641627023, −2.46004670940138431304770087729, −1.42223907089353263256010257066, 0,
1.42223907089353263256010257066, 2.46004670940138431304770087729, 3.31040685371487920512641627023, 4.09884680947828630682044141523, 5.09062225308872914362814185000, 5.86640160768240914318904531858, 6.34518743211658350505908566092, 7.29004645934603250674463684080, 7.83186370683019467591166338181
