| L(s) = 1 | + 7-s − 3·9-s + 20·37-s + 26·43-s − 6·49-s − 3·63-s + 22·67-s − 8·79-s + 9·81-s − 34·109-s + 22·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 23·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + ⋯ |
| L(s) = 1 | + 0.377·7-s − 9-s + 3.28·37-s + 3.96·43-s − 6/7·49-s − 0.377·63-s + 2.68·67-s − 0.900·79-s + 81-s − 3.25·109-s + 2·121-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 + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 1.76·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4410000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4410000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.495722479\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.495722479\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_2$ | \( 1 + p T^{2} \) |
| 5 | | \( 1 \) |
| 7 | $C_2$ | \( 1 - T + p T^{2} \) |
| good | 11 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 13 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 19 T + p T^{2} )( 1 + 19 T + p T^{2} ) \) |
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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
−9.392370219469317633615496673953, −8.959996768408671496646051101715, −8.357552991546846011215925798387, −8.212791670058767593650587445511, −7.71648291966824864121746920651, −7.60400890942188797442488002613, −6.79019970009611985461886590635, −6.71876485610259097606633427180, −5.85173560484428553557316107271, −5.80401584568793894548997609694, −5.58437134721207395232452429461, −4.77508339988735229769785415241, −4.31715088607608757116860871668, −4.23080757060767761464277536192, −3.41818524781695930131604683720, −2.97881154299357734548419582509, −2.34302864968474590573016515114, −2.23658816712640536133639029121, −1.08698661313618801790473441789, −0.65572117082124692658368154963,
0.65572117082124692658368154963, 1.08698661313618801790473441789, 2.23658816712640536133639029121, 2.34302864968474590573016515114, 2.97881154299357734548419582509, 3.41818524781695930131604683720, 4.23080757060767761464277536192, 4.31715088607608757116860871668, 4.77508339988735229769785415241, 5.58437134721207395232452429461, 5.80401584568793894548997609694, 5.85173560484428553557316107271, 6.71876485610259097606633427180, 6.79019970009611985461886590635, 7.60400890942188797442488002613, 7.71648291966824864121746920651, 8.212791670058767593650587445511, 8.357552991546846011215925798387, 8.959996768408671496646051101715, 9.392370219469317633615496673953
