| L(s) = 1 | + 7·2-s − 9·3-s + 33·4-s − 25·5-s − 63·6-s + 119·8-s + 81·9-s − 175·10-s − 297·12-s + 225·15-s + 305·16-s + 382·17-s + 567·18-s − 238·19-s − 825·20-s − 98·23-s − 1.07e3·24-s + 625·25-s − 729·27-s + 1.57e3·30-s − 1.91e3·31-s + 231·32-s + 2.67e3·34-s + 2.67e3·36-s − 1.66e3·38-s − 2.97e3·40-s − 2.02e3·45-s + ⋯ |
| L(s) = 1 | + 7/4·2-s − 3-s + 2.06·4-s − 5-s − 7/4·6-s + 1.85·8-s + 9-s − 7/4·10-s − 2.06·12-s + 15-s + 1.19·16-s + 1.32·17-s + 7/4·18-s − 0.659·19-s − 2.06·20-s − 0.185·23-s − 1.85·24-s + 25-s − 27-s + 7/4·30-s − 1.99·31-s + 0.225·32-s + 2.31·34-s + 2.06·36-s − 1.15·38-s − 1.85·40-s − 45-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.916729405\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.916729405\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + p^{2} T \) |
| 5 | \( 1 + p^{2} T \) |
| good | 2 | \( 1 - 7 T + p^{4} T^{2} \) |
| 7 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 11 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 13 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 17 | \( 1 - 382 T + p^{4} T^{2} \) |
| 19 | \( 1 + 238 T + p^{4} T^{2} \) |
| 23 | \( 1 + 98 T + p^{4} T^{2} \) |
| 29 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 31 | \( 1 + 1918 T + p^{4} T^{2} \) |
| 37 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 41 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 43 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 47 | \( 1 - 4222 T + p^{4} T^{2} \) |
| 53 | \( 1 + 1778 T + p^{4} T^{2} \) |
| 59 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 61 | \( 1 - 6482 T + p^{4} T^{2} \) |
| 67 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 71 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 73 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 79 | \( 1 + 2878 T + p^{4} T^{2} \) |
| 83 | \( 1 + 9938 T + p^{4} T^{2} \) |
| 89 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 97 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−18.78168838473308786979235308322, −16.71573762681553170214913763607, −15.71679296515720067809637440425, −14.56011136051617101079934785210, −12.80025688234506934326540519456, −11.97795017706656460998365633929, −10.83612731075073562657026873661, −7.22494531439750500171565665066, −5.54547990311578300861855697105, −3.96476322029534811803044665594,
3.96476322029534811803044665594, 5.54547990311578300861855697105, 7.22494531439750500171565665066, 10.83612731075073562657026873661, 11.97795017706656460998365633929, 12.80025688234506934326540519456, 14.56011136051617101079934785210, 15.71679296515720067809637440425, 16.71573762681553170214913763607, 18.78168838473308786979235308322
