| L(s) = 1 | + 9·3-s − 96·5-s + 212·7-s + 81·9-s + 668·11-s + 1.10e3·13-s − 864·15-s − 706·17-s − 604·19-s + 1.90e3·21-s + 1.06e3·23-s + 6.09e3·25-s + 729·27-s − 712·29-s − 364·31-s + 6.01e3·33-s − 2.03e4·35-s + 1.04e4·37-s + 9.97e3·39-s + 5.09e3·41-s − 2.14e4·43-s − 7.77e3·45-s + 1.71e4·47-s + 2.81e4·49-s − 6.35e3·51-s − 1.49e4·53-s − 6.41e4·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.71·5-s + 1.63·7-s + 1/3·9-s + 1.66·11-s + 1.81·13-s − 0.991·15-s − 0.592·17-s − 0.383·19-s + 0.944·21-s + 0.419·23-s + 1.94·25-s + 0.192·27-s − 0.157·29-s − 0.0680·31-s + 0.961·33-s − 2.80·35-s + 1.25·37-s + 1.04·39-s + 0.473·41-s − 1.77·43-s − 0.572·45-s + 1.13·47-s + 1.67·49-s − 0.342·51-s − 0.729·53-s − 2.85·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(3.233244767\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.233244767\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 + 96 T + p^{5} T^{2} \) |
| 7 | \( 1 - 212 T + p^{5} T^{2} \) |
| 11 | \( 1 - 668 T + p^{5} T^{2} \) |
| 13 | \( 1 - 1108 T + p^{5} T^{2} \) |
| 17 | \( 1 + 706 T + p^{5} T^{2} \) |
| 19 | \( 1 + 604 T + p^{5} T^{2} \) |
| 23 | \( 1 - 1064 T + p^{5} T^{2} \) |
| 29 | \( 1 + 712 T + p^{5} T^{2} \) |
| 31 | \( 1 + 364 T + p^{5} T^{2} \) |
| 37 | \( 1 - 10412 T + p^{5} T^{2} \) |
| 41 | \( 1 - 5094 T + p^{5} T^{2} \) |
| 43 | \( 1 + 21476 T + p^{5} T^{2} \) |
| 47 | \( 1 - 17192 T + p^{5} T^{2} \) |
| 53 | \( 1 + 14920 T + p^{5} T^{2} \) |
| 59 | \( 1 + 32724 T + p^{5} T^{2} \) |
| 61 | \( 1 + 21060 T + p^{5} T^{2} \) |
| 67 | \( 1 - 5268 T + p^{5} T^{2} \) |
| 71 | \( 1 - 21208 T + p^{5} T^{2} \) |
| 73 | \( 1 + 3174 T + p^{5} T^{2} \) |
| 79 | \( 1 - 55316 T + p^{5} T^{2} \) |
| 83 | \( 1 + 96476 T + p^{5} T^{2} \) |
| 89 | \( 1 + 67494 T + p^{5} T^{2} \) |
| 97 | \( 1 - 958 p T + p^{5} T^{2} \) |
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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
−9.127486291005481179622391242967, −8.550916885631252570488288736814, −8.063089145260976937265046777582, −7.19889626019616606151785186214, −6.22542599464881554225884225460, −4.59700328965388720496362282162, −4.12578105040825842043262231221, −3.35887135735575958132971278661, −1.67307603205268508831636961759, −0.894158529568006891263153085926,
0.894158529568006891263153085926, 1.67307603205268508831636961759, 3.35887135735575958132971278661, 4.12578105040825842043262231221, 4.59700328965388720496362282162, 6.22542599464881554225884225460, 7.19889626019616606151785186214, 8.063089145260976937265046777582, 8.550916885631252570488288736814, 9.127486291005481179622391242967
