| L(s) = 1 | − 2-s + 4-s − 8-s − 2·9-s + 8·13-s + 16-s − 12·17-s + 2·18-s − 10·25-s − 8·26-s − 12·29-s − 32-s + 12·34-s − 2·36-s + 4·37-s − 12·41-s + 10·50-s + 8·52-s + 12·53-s + 12·58-s − 16·61-s + 64-s − 12·68-s + 2·72-s − 4·73-s − 4·74-s − 5·81-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.353·8-s − 2/3·9-s + 2.21·13-s + 1/4·16-s − 2.91·17-s + 0.471·18-s − 2·25-s − 1.56·26-s − 2.22·29-s − 0.176·32-s + 2.05·34-s − 1/3·36-s + 0.657·37-s − 1.87·41-s + 1.41·50-s + 1.10·52-s + 1.64·53-s + 1.57·58-s − 2.04·61-s + 1/8·64-s − 1.45·68-s + 0.235·72-s − 0.468·73-s − 0.464·74-s − 5/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76832 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76832 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( 1 + T \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 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
−9.205467104844532229657218112988, −9.044637307272642732451675861332, −8.624667079434487291694904422005, −8.199638847074308569053560326614, −7.55563465371540871894331307120, −7.01851678171449461328017406897, −6.24500065333065006116855945568, −6.12986038892916091786195943044, −5.53331414913354333592453982899, −4.55372588498345584946220609716, −3.87985229263682105648190826671, −3.42554077807776045337947490220, −2.25140513775369021989127014661, −1.73289286183865826039844070287, 0,
1.73289286183865826039844070287, 2.25140513775369021989127014661, 3.42554077807776045337947490220, 3.87985229263682105648190826671, 4.55372588498345584946220609716, 5.53331414913354333592453982899, 6.12986038892916091786195943044, 6.24500065333065006116855945568, 7.01851678171449461328017406897, 7.55563465371540871894331307120, 8.199638847074308569053560326614, 8.624667079434487291694904422005, 9.044637307272642732451675861332, 9.205467104844532229657218112988
