| L(s) = 1 | + 3·5-s − 2·7-s − 6·11-s − 5·13-s + 6·17-s + 4·19-s + 6·23-s + 5·25-s + 3·29-s + 4·31-s − 6·35-s + 10·37-s − 6·41-s + 10·43-s + 7·49-s + 12·53-s − 18·55-s − 12·59-s − 5·61-s − 15·65-s − 2·67-s − 12·71-s − 2·73-s + 12·77-s + 10·79-s + 18·85-s + 6·89-s + ⋯ |
| L(s) = 1 | + 1.34·5-s − 0.755·7-s − 1.80·11-s − 1.38·13-s + 1.45·17-s + 0.917·19-s + 1.25·23-s + 25-s + 0.557·29-s + 0.718·31-s − 1.01·35-s + 1.64·37-s − 0.937·41-s + 1.52·43-s + 49-s + 1.64·53-s − 2.42·55-s − 1.56·59-s − 0.640·61-s − 1.86·65-s − 0.244·67-s − 1.42·71-s − 0.234·73-s + 1.36·77-s + 1.12·79-s + 1.95·85-s + 0.635·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 104976 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 104976 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.647586677\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.647586677\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 - 3 T + 4 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 2 T - 3 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 6 T + 25 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 6 T + 13 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 3 T - 20 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 6 T - 5 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 10 T + 57 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 12 T + 85 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 5 T - 36 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 2 T - 63 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 10 T + 21 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 10 T + 3 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
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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
−11.96297238117988667640298414736, −11.41470032453232554750095195641, −10.53374388284959934229764616553, −10.37864646771819563033577462290, −10.11664526213416518522555434750, −9.494133237303159615432348337833, −9.323598198158614980138775093847, −8.698233963198732900725570156511, −7.72041112268156637662184917747, −7.70411241119005063996128701604, −7.14427200571498360767566607545, −6.40285866834164375620609452158, −5.86595632836788444094026650577, −5.40080429617027325624458101214, −5.06591642150914633137430871306, −4.42296694578054834532579966763, −3.13604724997641892669080831300, −2.84814439494514772339290462718, −2.31321220030172474812161336887, −0.957563683851807472073539154453,
0.957563683851807472073539154453, 2.31321220030172474812161336887, 2.84814439494514772339290462718, 3.13604724997641892669080831300, 4.42296694578054834532579966763, 5.06591642150914633137430871306, 5.40080429617027325624458101214, 5.86595632836788444094026650577, 6.40285866834164375620609452158, 7.14427200571498360767566607545, 7.70411241119005063996128701604, 7.72041112268156637662184917747, 8.698233963198732900725570156511, 9.323598198158614980138775093847, 9.494133237303159615432348337833, 10.11664526213416518522555434750, 10.37864646771819563033577462290, 10.53374388284959934229764616553, 11.41470032453232554750095195641, 11.96297238117988667640298414736
