| L(s) = 1 | + 2-s + 6·5-s − 5·7-s − 8-s − 3·9-s + 6·10-s − 12·11-s + 5·13-s − 5·14-s − 16-s − 3·18-s + 10·19-s − 12·22-s − 6·23-s + 19·25-s + 5·26-s + 12·29-s + 8·31-s − 30·35-s + 15·37-s + 10·38-s − 6·40-s − 18·41-s − 43-s − 18·45-s − 6·46-s + 12·47-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 2.68·5-s − 1.88·7-s − 0.353·8-s − 9-s + 1.89·10-s − 3.61·11-s + 1.38·13-s − 1.33·14-s − 1/4·16-s − 0.707·18-s + 2.29·19-s − 2.55·22-s − 1.25·23-s + 19/5·25-s + 0.980·26-s + 2.22·29-s + 1.43·31-s − 5.07·35-s + 2.46·37-s + 1.62·38-s − 0.948·40-s − 2.81·41-s − 0.152·43-s − 2.68·45-s − 0.884·46-s + 1.75·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 298116 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 298116 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.516778032\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.516778032\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + T^{2} \) |
| 3 | $C_2$ | \( 1 + p T^{2} \) |
| 7 | $C_2$ | \( 1 + 5 T + p T^{2} \) |
| 13 | $C_2$ | \( 1 - 5 T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - 6 T + 17 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 6 T + 35 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 12 T + 77 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 8 T + 33 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 15 T + 112 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 18 T + 149 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + T - 42 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 12 T + 95 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 6 T + 65 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 95 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 8 T - 15 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 154 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 6 T + 101 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 7 T - 48 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 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
−10.76805866182085146390535990006, −10.27852748917222910848548541681, −10.13064444449649753280353087915, −9.850134957052111581445986083181, −9.619674088930535052958322922042, −8.715468787332783618610743762956, −8.611052073188345314422690095327, −7.87408698039281699304560761933, −7.45389015000565322221638444024, −6.39400795127844896165571574821, −6.30927956356257124968884540011, −5.79011466757763464378975471089, −5.73210287656153197836196568733, −5.01363187447166809691319042922, −4.87073791544635653330483103059, −3.41842560595847320580206263647, −3.00467342949300418278402717611, −2.65687199739547416659753892203, −2.31474582188167809097449907949, −0.803005845683041738013980557988,
0.803005845683041738013980557988, 2.31474582188167809097449907949, 2.65687199739547416659753892203, 3.00467342949300418278402717611, 3.41842560595847320580206263647, 4.87073791544635653330483103059, 5.01363187447166809691319042922, 5.73210287656153197836196568733, 5.79011466757763464378975471089, 6.30927956356257124968884540011, 6.39400795127844896165571574821, 7.45389015000565322221638444024, 7.87408698039281699304560761933, 8.611052073188345314422690095327, 8.715468787332783618610743762956, 9.619674088930535052958322922042, 9.850134957052111581445986083181, 10.13064444449649753280353087915, 10.27852748917222910848548541681, 10.76805866182085146390535990006
