| L(s) = 1 | − 2·2-s + 3·4-s − 4·8-s − 6·9-s + 2·11-s + 5·16-s + 12·18-s − 4·22-s − 16·23-s − 2·25-s − 12·29-s − 6·32-s − 18·36-s − 12·37-s − 8·43-s + 6·44-s + 32·46-s + 4·50-s + 12·53-s + 24·58-s + 7·64-s − 8·67-s + 24·72-s + 24·74-s + 27·81-s + 16·86-s − 8·88-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 3/2·4-s − 1.41·8-s − 2·9-s + 0.603·11-s + 5/4·16-s + 2.82·18-s − 0.852·22-s − 3.33·23-s − 2/5·25-s − 2.22·29-s − 1.06·32-s − 3·36-s − 1.97·37-s − 1.21·43-s + 0.904·44-s + 4.71·46-s + 0.565·50-s + 1.64·53-s + 3.15·58-s + 7/8·64-s − 0.977·67-s + 2.82·72-s + 2.78·74-s + 3·81-s + 1.72·86-s − 0.852·88-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1162084 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1162084 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 7 | | \( 1 \) |
| 11 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 3 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 5 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 86 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 86 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 90 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 74 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 158 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 66 T^{2} + p^{2} T^{4} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.530730185409156455610738588161, −9.283237433028884152800309579630, −8.734877997248979398883447111660, −8.583415091695541155962191425043, −8.039633612488981166741343372411, −7.88513291085923387526654283203, −7.30890581613593169430129609524, −6.85101474999731051432473448282, −6.17053049575869876704436871049, −6.09707809914066463630119882667, −5.42267652915547103492609807415, −5.35982384699957127080629626720, −4.18712210120360699910405087214, −3.65576217832408950156496650701, −3.40211026996790954756177269831, −2.46348138242626626833794123960, −2.08526057766359163412999320943, −1.53936938227648347338790331411, 0, 0,
1.53936938227648347338790331411, 2.08526057766359163412999320943, 2.46348138242626626833794123960, 3.40211026996790954756177269831, 3.65576217832408950156496650701, 4.18712210120360699910405087214, 5.35982384699957127080629626720, 5.42267652915547103492609807415, 6.09707809914066463630119882667, 6.17053049575869876704436871049, 6.85101474999731051432473448282, 7.30890581613593169430129609524, 7.88513291085923387526654283203, 8.039633612488981166741343372411, 8.583415091695541155962191425043, 8.734877997248979398883447111660, 9.283237433028884152800309579630, 9.530730185409156455610738588161
