| L(s) = 1 | + 5·2-s + 3·3-s + 17·4-s + 15·6-s + 3·7-s + 45·8-s + 9·9-s − 11·11-s + 51·12-s + 32·13-s + 15·14-s + 89·16-s + 33·17-s + 45·18-s + 47·19-s + 9·21-s − 55·22-s + 113·23-s + 135·24-s + 160·26-s + 27·27-s + 51·28-s − 54·29-s + 178·31-s + 85·32-s − 33·33-s + 165·34-s + ⋯ |
| L(s) = 1 | + 1.76·2-s + 0.577·3-s + 17/8·4-s + 1.02·6-s + 0.161·7-s + 1.98·8-s + 1/3·9-s − 0.301·11-s + 1.22·12-s + 0.682·13-s + 0.286·14-s + 1.39·16-s + 0.470·17-s + 0.589·18-s + 0.567·19-s + 0.0935·21-s − 0.533·22-s + 1.02·23-s + 1.14·24-s + 1.20·26-s + 0.192·27-s + 0.344·28-s − 0.345·29-s + 1.03·31-s + 0.469·32-s − 0.174·33-s + 0.832·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(8.318744885\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.318744885\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - p T \) |
| 5 | \( 1 \) |
| 11 | \( 1 + p T \) |
| good | 2 | \( 1 - 5 T + p^{3} T^{2} \) |
| 7 | \( 1 - 3 T + p^{3} T^{2} \) |
| 13 | \( 1 - 32 T + p^{3} T^{2} \) |
| 17 | \( 1 - 33 T + p^{3} T^{2} \) |
| 19 | \( 1 - 47 T + p^{3} T^{2} \) |
| 23 | \( 1 - 113 T + p^{3} T^{2} \) |
| 29 | \( 1 + 54 T + p^{3} T^{2} \) |
| 31 | \( 1 - 178 T + p^{3} T^{2} \) |
| 37 | \( 1 - 19 T + p^{3} T^{2} \) |
| 41 | \( 1 - 139 T + p^{3} T^{2} \) |
| 43 | \( 1 + 308 T + p^{3} T^{2} \) |
| 47 | \( 1 - 195 T + p^{3} T^{2} \) |
| 53 | \( 1 - 152 T + p^{3} T^{2} \) |
| 59 | \( 1 + 625 T + p^{3} T^{2} \) |
| 61 | \( 1 - 320 T + p^{3} T^{2} \) |
| 67 | \( 1 - 200 T + p^{3} T^{2} \) |
| 71 | \( 1 + 947 T + p^{3} T^{2} \) |
| 73 | \( 1 + 448 T + p^{3} T^{2} \) |
| 79 | \( 1 + 721 T + p^{3} T^{2} \) |
| 83 | \( 1 - 142 T + p^{3} T^{2} \) |
| 89 | \( 1 - 404 T + p^{3} T^{2} \) |
| 97 | \( 1 - 79 T + p^{3} T^{2} \) |
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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
−10.04390129766390676940889069135, −8.901946354398418641113449894813, −7.86790127303902284136801313811, −7.04136010106577222496411036181, −6.11401248338777494989989463847, −5.23485140533083100872796031018, −4.39957337738952996557997575280, −3.40427502817067675593100719523, −2.72383329691811996825335001552, −1.40673720782226677466309288138,
1.40673720782226677466309288138, 2.72383329691811996825335001552, 3.40427502817067675593100719523, 4.39957337738952996557997575280, 5.23485140533083100872796031018, 6.11401248338777494989989463847, 7.04136010106577222496411036181, 7.86790127303902284136801313811, 8.901946354398418641113449894813, 10.04390129766390676940889069135
