| L(s) = 1 | + 2·2-s − 3-s + 3·4-s − 2·6-s + 6·7-s + 4·8-s − 2·9-s − 3·12-s + 12·14-s + 5·16-s − 4·18-s − 19-s − 6·21-s − 4·24-s + 6·25-s + 5·27-s + 18·28-s − 10·29-s + 6·32-s − 6·36-s − 2·38-s − 16·41-s − 12·42-s + 8·43-s − 5·48-s + 13·49-s + 12·50-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 0.577·3-s + 3/2·4-s − 0.816·6-s + 2.26·7-s + 1.41·8-s − 2/3·9-s − 0.866·12-s + 3.20·14-s + 5/4·16-s − 0.942·18-s − 0.229·19-s − 1.30·21-s − 0.816·24-s + 6/5·25-s + 0.962·27-s + 3.40·28-s − 1.85·29-s + 1.06·32-s − 36-s − 0.324·38-s − 2.49·41-s − 1.85·42-s + 1.21·43-s − 0.721·48-s + 13/7·49-s + 1.69·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 246924 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 246924 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.444888249\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.444888249\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 3 | $C_2$ | \( 1 + T + p T^{2} \) |
| 19 | $C_1$ | \( 1 + T \) |
| good | 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 15 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
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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
−8.602687591287036982918634833707, −8.408486909011940238981299566497, −8.066304209751900883680372833625, −7.25038064772684706053554512263, −7.08630692294978510791408292834, −6.40001715187249414844342874557, −5.83278378320473922139761712408, −5.28069074518443602737935578596, −4.97804072056498150514618992266, −4.85908639596030014967981752307, −3.86057015395193111701613139713, −3.61504607409621719549683932368, −2.49657072322899130135959102215, −2.06231205992969271621001293694, −1.18389774049284853047939869594,
1.18389774049284853047939869594, 2.06231205992969271621001293694, 2.49657072322899130135959102215, 3.61504607409621719549683932368, 3.86057015395193111701613139713, 4.85908639596030014967981752307, 4.97804072056498150514618992266, 5.28069074518443602737935578596, 5.83278378320473922139761712408, 6.40001715187249414844342874557, 7.08630692294978510791408292834, 7.25038064772684706053554512263, 8.066304209751900883680372833625, 8.408486909011940238981299566497, 8.602687591287036982918634833707
