| L(s) = 1 | − 2·4-s − 5-s + 2·7-s − 3·11-s − 4·13-s + 4·16-s − 6·17-s − 19-s + 2·20-s − 6·23-s + 25-s − 4·28-s − 9·29-s − 31-s − 2·35-s + 8·37-s + 3·41-s − 4·43-s + 6·44-s + 12·47-s − 3·49-s + 8·52-s + 6·53-s + 3·55-s + 3·59-s − 10·61-s − 8·64-s + ⋯ |
| L(s) = 1 | − 4-s − 0.447·5-s + 0.755·7-s − 0.904·11-s − 1.10·13-s + 16-s − 1.45·17-s − 0.229·19-s + 0.447·20-s − 1.25·23-s + 1/5·25-s − 0.755·28-s − 1.67·29-s − 0.179·31-s − 0.338·35-s + 1.31·37-s + 0.468·41-s − 0.609·43-s + 0.904·44-s + 1.75·47-s − 3/7·49-s + 1.10·52-s + 0.824·53-s + 0.404·55-s + 0.390·59-s − 1.28·61-s − 64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| good | 2 | \( 1 + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 + T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 14 T + p T^{2} \) |
| 71 | \( 1 + 3 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 15 T + p T^{2} \) |
| 97 | \( 1 + 4 T + p T^{2} \) |
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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
−10.80093468211168061044076040418, −9.842969236122360080923884113349, −8.922941609669290152495843868069, −8.027105501047314182413259178583, −7.36532580760380206733382868882, −5.75188631797463493014423176019, −4.76552433542106688375778775229, −4.04017076909632160868815003758, −2.30189805881882976209364768483, 0,
2.30189805881882976209364768483, 4.04017076909632160868815003758, 4.76552433542106688375778775229, 5.75188631797463493014423176019, 7.36532580760380206733382868882, 8.027105501047314182413259178583, 8.922941609669290152495843868069, 9.842969236122360080923884113349, 10.80093468211168061044076040418
