| L(s) = 1 | − 3-s − 7·5-s + 20·7-s + 27·9-s + 35·11-s + 132·13-s + 7·15-s − 59·17-s + 137·19-s − 20·21-s − 7·23-s + 125·25-s − 80·27-s + 212·29-s + 75·31-s − 35·33-s − 140·35-s − 11·37-s − 132·39-s − 996·41-s − 520·43-s − 189·45-s − 171·47-s + 57·49-s + 59·51-s + 417·53-s − 245·55-s + ⋯ |
| L(s) = 1 | − 0.192·3-s − 0.626·5-s + 1.07·7-s + 9-s + 0.959·11-s + 2.81·13-s + 0.120·15-s − 0.841·17-s + 1.65·19-s − 0.207·21-s − 0.0634·23-s + 25-s − 0.570·27-s + 1.35·29-s + 0.434·31-s − 0.184·33-s − 0.676·35-s − 0.0488·37-s − 0.541·39-s − 3.79·41-s − 1.84·43-s − 0.626·45-s − 0.530·47-s + 0.166·49-s + 0.161·51-s + 1.08·53-s − 0.600·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12544 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12544 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.753713960\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.753713960\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 7 | $C_2$ | \( 1 - 20 T + p^{3} T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + T - 26 T^{2} + p^{3} T^{3} + p^{6} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 7 T - 76 T^{2} + 7 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 35 T - 106 T^{2} - 35 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 66 T + p^{3} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 59 T - 1432 T^{2} + 59 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 137 T + 11910 T^{2} - 137 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 7 T - 12118 T^{2} + 7 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 106 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 75 T - 24166 T^{2} - 75 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 11 T - 50532 T^{2} + 11 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 498 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 260 T + p^{3} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 171 T - 74582 T^{2} + 171 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 417 T + 25012 T^{2} - 417 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 17 T - 205090 T^{2} + 17 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 51 T - 224380 T^{2} + 51 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 439 T - 108042 T^{2} - 439 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 784 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 295 T - 301992 T^{2} + 295 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 495 T - 248014 T^{2} + 495 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 932 T + p^{3} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 873 T + 57160 T^{2} - 873 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 290 T + p^{3} T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.24831628589291895815022951191, −13.20216117057822084934598657134, −12.07160688257089399611132425111, −11.80150690184479113187725869763, −11.25085703868421261012768432946, −11.13613888674052675324148817839, −10.14349508903992616982421448462, −9.943992475412367864589102768047, −8.749883203989109086502694423882, −8.551759326689261201310840605209, −8.212789062632414080876327895333, −7.21086197163421905529926249460, −6.69326887992141851398920707387, −6.28330267556535683809957265939, −5.14072807366062122748080176405, −4.72235878389007995424466189618, −3.74014406165420185107643424478, −3.44622426064655006654250058404, −1.54602903899650683507618455726, −1.14992680721516712133275645612,
1.14992680721516712133275645612, 1.54602903899650683507618455726, 3.44622426064655006654250058404, 3.74014406165420185107643424478, 4.72235878389007995424466189618, 5.14072807366062122748080176405, 6.28330267556535683809957265939, 6.69326887992141851398920707387, 7.21086197163421905529926249460, 8.212789062632414080876327895333, 8.551759326689261201310840605209, 8.749883203989109086502694423882, 9.943992475412367864589102768047, 10.14349508903992616982421448462, 11.13613888674052675324148817839, 11.25085703868421261012768432946, 11.80150690184479113187725869763, 12.07160688257089399611132425111, 13.20216117057822084934598657134, 13.24831628589291895815022951191
