| L(s) = 1 | − 2·2-s + 3-s + 2·4-s − 2·6-s + 7-s − 5·9-s + 2·12-s + 4·13-s − 2·14-s − 4·16-s − 8·17-s + 10·18-s + 21-s + 2·25-s − 8·26-s − 8·27-s + 2·28-s + 5·31-s + 8·32-s + 16·34-s − 10·36-s + 4·39-s − 2·42-s − 4·48-s − 6·49-s − 4·50-s − 8·51-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 0.577·3-s + 4-s − 0.816·6-s + 0.377·7-s − 5/3·9-s + 0.577·12-s + 1.10·13-s − 0.534·14-s − 16-s − 1.94·17-s + 2.35·18-s + 0.218·21-s + 2/5·25-s − 1.56·26-s − 1.53·27-s + 0.377·28-s + 0.898·31-s + 1.41·32-s + 2.74·34-s − 5/3·36-s + 0.640·39-s − 0.308·42-s − 0.577·48-s − 6/7·49-s − 0.565·50-s − 1.12·51-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 376712 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 376712 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + p T + p T^{2} \) |
| 7 | $C_2$ | \( 1 - T + p T^{2} \) |
| 31 | $C_2$ | \( 1 - 5 T + p T^{2} \) |
| good | 3 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + p T^{2} ) \) |
| 5 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 + 3 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 27 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 32 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 67 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 12 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 81 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 76 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 + 85 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
| 97 | $C_2^2$ | \( 1 + 95 T^{2} + p^{2} T^{4} \) |
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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
−8.481765104704243458107467924444, −8.163472940480207093872877495297, −7.988581065001002007561354155485, −7.18846052961559921650470126901, −6.63868303594132412203305196171, −6.36847538569758616794688200460, −5.78491875162769036477219955343, −5.07380934291241382770792854721, −4.60327840401235859320218368645, −3.87390136361321605644092997303, −3.26777674709577206100071927561, −2.43454301063639847481232157013, −2.18728853203759025510143644775, −1.13036261040133569285579328156, 0,
1.13036261040133569285579328156, 2.18728853203759025510143644775, 2.43454301063639847481232157013, 3.26777674709577206100071927561, 3.87390136361321605644092997303, 4.60327840401235859320218368645, 5.07380934291241382770792854721, 5.78491875162769036477219955343, 6.36847538569758616794688200460, 6.63868303594132412203305196171, 7.18846052961559921650470126901, 7.988581065001002007561354155485, 8.163472940480207093872877495297, 8.481765104704243458107467924444
