| L(s) = 1 | + 3·2-s + 4·4-s + 5-s − 5·7-s + 3·8-s − 3·9-s + 3·10-s + 6·11-s − 15·14-s + 3·16-s + 6·17-s − 9·18-s − 12·19-s + 4·20-s + 18·22-s − 3·23-s − 20·28-s − 6·31-s + 6·32-s + 18·34-s − 5·35-s − 12·36-s − 4·37-s − 36·38-s + 3·40-s + 6·41-s + 2·43-s + ⋯ |
| L(s) = 1 | + 2.12·2-s + 2·4-s + 0.447·5-s − 1.88·7-s + 1.06·8-s − 9-s + 0.948·10-s + 1.80·11-s − 4.00·14-s + 3/4·16-s + 1.45·17-s − 2.12·18-s − 2.75·19-s + 0.894·20-s + 3.83·22-s − 0.625·23-s − 3.77·28-s − 1.07·31-s + 1.06·32-s + 3.08·34-s − 0.845·35-s − 2·36-s − 0.657·37-s − 5.83·38-s + 0.474·40-s + 0.937·41-s + 0.304·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11025 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.401869050\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.401869050\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_2$ | \( 1 + p T^{2} \) |
| 5 | $C_2$ | \( 1 - T + T^{2} \) |
| 7 | $C_2$ | \( 1 + 5 T + p T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - 3 T + 5 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 6 T + 23 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 6 T + 19 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 12 T + 67 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 3 T + 26 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 55 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 6 T + 43 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 4 T - 21 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 9 T + 88 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 13 T + 102 T^{2} - 13 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 94 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 6 T + 85 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 16 T + 177 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 3 T - 80 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 86 T^{2} + p^{2} T^{4} \) |
| 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
−14.16221337083112004545461272380, −13.40234493524462221707469713664, −13.16528479108188247367854058641, −12.47457273420476726697474355545, −12.17778866930277049334532663122, −12.07714230436148351318471699109, −10.78827111660785066695676928056, −10.71486310466920217838917030012, −9.638832843579580143076439947509, −9.371376049804196766070883257826, −8.713229594752641254721866506392, −7.88175014359809323304505485828, −6.64618256560601271719208051527, −6.51566663884681990143475985179, −5.88733815386282041729798579625, −5.53579999814029133993583250022, −4.43470446750789562720769364405, −3.61524603998433737121217567054, −3.59483606009000309654685246745, −2.35185031018607232999486137825,
2.35185031018607232999486137825, 3.59483606009000309654685246745, 3.61524603998433737121217567054, 4.43470446750789562720769364405, 5.53579999814029133993583250022, 5.88733815386282041729798579625, 6.51566663884681990143475985179, 6.64618256560601271719208051527, 7.88175014359809323304505485828, 8.713229594752641254721866506392, 9.371376049804196766070883257826, 9.638832843579580143076439947509, 10.71486310466920217838917030012, 10.78827111660785066695676928056, 12.07714230436148351318471699109, 12.17778866930277049334532663122, 12.47457273420476726697474355545, 13.16528479108188247367854058641, 13.40234493524462221707469713664, 14.16221337083112004545461272380
