| L(s) = 1 | − 9-s − 8·13-s + 6·17-s − 8·29-s − 16·37-s + 10·41-s − 14·49-s − 8·53-s − 16·61-s + 18·73-s − 8·81-s + 30·89-s + 4·97-s + 2·113-s + 8·117-s − 17·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 6·153-s + 157-s + 163-s + 167-s + 22·169-s + ⋯ |
| L(s) = 1 | − 1/3·9-s − 2.21·13-s + 1.45·17-s − 1.48·29-s − 2.63·37-s + 1.56·41-s − 2·49-s − 1.09·53-s − 2.04·61-s + 2.10·73-s − 8/9·81-s + 3.17·89-s + 0.406·97-s + 0.188·113-s + 0.739·117-s − 1.54·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s − 0.485·153-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 1.69·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 40960000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40960000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 + 17 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 + 33 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 89 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 121 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 15 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.62630228742111104405410194476, −7.59153382329151969011892383272, −7.31567032450017808789058034206, −6.86410207211914380742488653618, −6.42579197955964272677006762392, −6.06164623913289272520590634004, −5.71939879708007336073367290940, −5.20261900503575326851900162906, −4.98615117825145690684154539801, −4.84454978449271398229762964781, −4.26275178001913604167947144573, −3.66526982431319312576559928492, −3.26838303956153138733833934433, −3.20993716958643536365600947645, −2.40042801312707350751029567853, −2.18021503424833757021371908691, −1.63557109521418909610674569274, −1.07271981202458125192992433112, 0, 0,
1.07271981202458125192992433112, 1.63557109521418909610674569274, 2.18021503424833757021371908691, 2.40042801312707350751029567853, 3.20993716958643536365600947645, 3.26838303956153138733833934433, 3.66526982431319312576559928492, 4.26275178001913604167947144573, 4.84454978449271398229762964781, 4.98615117825145690684154539801, 5.20261900503575326851900162906, 5.71939879708007336073367290940, 6.06164623913289272520590634004, 6.42579197955964272677006762392, 6.86410207211914380742488653618, 7.31567032450017808789058034206, 7.59153382329151969011892383272, 7.62630228742111104405410194476
