| L(s) = 1 | + 9·3-s + 38·5-s + 120·7-s + 81·9-s + 524·11-s − 962·13-s + 342·15-s − 1.35e3·17-s − 2.28e3·19-s + 1.08e3·21-s + 2.55e3·23-s − 1.68e3·25-s + 729·27-s + 3.96e3·29-s − 2.99e3·31-s + 4.71e3·33-s + 4.56e3·35-s + 1.32e4·37-s − 8.65e3·39-s − 1.51e4·41-s − 7.31e3·43-s + 3.07e3·45-s − 6.96e3·47-s − 2.40e3·49-s − 1.22e4·51-s − 1.74e4·53-s + 1.99e4·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.679·5-s + 0.925·7-s + 1/3·9-s + 1.30·11-s − 1.57·13-s + 0.392·15-s − 1.13·17-s − 1.45·19-s + 0.534·21-s + 1.00·23-s − 0.537·25-s + 0.192·27-s + 0.875·29-s − 0.559·31-s + 0.753·33-s + 0.629·35-s + 1.58·37-s − 0.911·39-s − 1.40·41-s − 0.603·43-s + 0.226·45-s − 0.459·47-s − 0.143·49-s − 0.657·51-s − 0.854·53-s + 0.887·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 24 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 24 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.928987107\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.928987107\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - p^{2} T \) |
| good | 5 | \( 1 - 38 T + p^{5} T^{2} \) |
| 7 | \( 1 - 120 T + p^{5} T^{2} \) |
| 11 | \( 1 - 524 T + p^{5} T^{2} \) |
| 13 | \( 1 + 74 p T + p^{5} T^{2} \) |
| 17 | \( 1 + 1358 T + p^{5} T^{2} \) |
| 19 | \( 1 + 2284 T + p^{5} T^{2} \) |
| 23 | \( 1 - 2552 T + p^{5} T^{2} \) |
| 29 | \( 1 - 3966 T + p^{5} T^{2} \) |
| 31 | \( 1 + 2992 T + p^{5} T^{2} \) |
| 37 | \( 1 - 13206 T + p^{5} T^{2} \) |
| 41 | \( 1 + 15126 T + p^{5} T^{2} \) |
| 43 | \( 1 + 7316 T + p^{5} T^{2} \) |
| 47 | \( 1 + 6960 T + p^{5} T^{2} \) |
| 53 | \( 1 + 17482 T + p^{5} T^{2} \) |
| 59 | \( 1 - 33884 T + p^{5} T^{2} \) |
| 61 | \( 1 - 39118 T + p^{5} T^{2} \) |
| 67 | \( 1 - 32996 T + p^{5} T^{2} \) |
| 71 | \( 1 - 14248 T + p^{5} T^{2} \) |
| 73 | \( 1 + 35990 T + p^{5} T^{2} \) |
| 79 | \( 1 + 29888 T + p^{5} T^{2} \) |
| 83 | \( 1 + 51884 T + p^{5} T^{2} \) |
| 89 | \( 1 - 30714 T + p^{5} T^{2} \) |
| 97 | \( 1 + 48478 T + p^{5} T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−16.99934901040862024042755054926, −14.97927458831224075147743346203, −14.37903269208825237096795687606, −12.98084546790580910875610888961, −11.43763690437101317304439523025, −9.785558897273776632932303567427, −8.544402023219668295329327693927, −6.77658823472690597792837142628, −4.60506937795549603601243275468, −2.05499535456184539093809832567,
2.05499535456184539093809832567, 4.60506937795549603601243275468, 6.77658823472690597792837142628, 8.544402023219668295329327693927, 9.785558897273776632932303567427, 11.43763690437101317304439523025, 12.98084546790580910875610888961, 14.37903269208825237096795687606, 14.97927458831224075147743346203, 16.99934901040862024042755054926
