| L(s) = 1 | − 2·5-s + 4·11-s − 6·13-s − 6·17-s − 12·23-s − 7·25-s + 6·37-s + 5·49-s − 8·55-s + 20·59-s + 12·65-s + 24·67-s − 32·83-s + 12·85-s − 8·103-s + 30·109-s + 12·113-s + 24·115-s − 10·121-s + 26·125-s + 127-s + 131-s + 137-s + 139-s − 24·143-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 1.20·11-s − 1.66·13-s − 1.45·17-s − 2.50·23-s − 7/5·25-s + 0.986·37-s + 5/7·49-s − 1.07·55-s + 2.60·59-s + 1.48·65-s + 2.93·67-s − 3.51·83-s + 1.30·85-s − 0.788·103-s + 2.87·109-s + 1.12·113-s + 2.23·115-s − 0.909·121-s + 2.32·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 2.00·143-s + 0.0819·149-s + 0.0813·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14017536 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14017536 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3007719851\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3007719851\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 13 | $C_2$ | \( 1 + 6 T + p T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 45 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 117 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 162 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
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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
−8.521627997179533620272873888961, −8.350174052216300662525422634468, −7.971741426722753366922585104685, −7.62046902975398087474564107542, −7.22895213128669574766760036898, −6.82162951728595368314451217572, −6.69206385819191479688383533557, −6.03935617624287427887833445924, −5.77697655521623084653260108503, −5.39696187218225109748207916278, −4.74963001396955454767547963126, −4.33348688716486543184726182852, −4.12096496599099372486847831525, −3.82971133827768777415195765379, −3.42878837016600000907123002105, −2.48037220080733794914316177378, −2.27692366594132568685905235280, −1.97353877672884316303680034318, −1.05229788046133669700911800195, −0.17471963942233941914089943808,
0.17471963942233941914089943808, 1.05229788046133669700911800195, 1.97353877672884316303680034318, 2.27692366594132568685905235280, 2.48037220080733794914316177378, 3.42878837016600000907123002105, 3.82971133827768777415195765379, 4.12096496599099372486847831525, 4.33348688716486543184726182852, 4.74963001396955454767547963126, 5.39696187218225109748207916278, 5.77697655521623084653260108503, 6.03935617624287427887833445924, 6.69206385819191479688383533557, 6.82162951728595368314451217572, 7.22895213128669574766760036898, 7.62046902975398087474564107542, 7.971741426722753366922585104685, 8.350174052216300662525422634468, 8.521627997179533620272873888961
