| L(s) = 1 | + 2-s + 3-s − 2·5-s + 6-s + 2·7-s − 8-s − 2·10-s + 11-s − 2·13-s + 2·14-s − 2·15-s − 16-s − 5·17-s − 5·19-s + 2·21-s + 22-s + 8·23-s − 24-s − 7·25-s − 2·26-s − 27-s + 2·29-s − 2·30-s + 6·31-s + 33-s − 5·34-s − 4·35-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s − 0.894·5-s + 0.408·6-s + 0.755·7-s − 0.353·8-s − 0.632·10-s + 0.301·11-s − 0.554·13-s + 0.534·14-s − 0.516·15-s − 1/4·16-s − 1.21·17-s − 1.14·19-s + 0.436·21-s + 0.213·22-s + 1.66·23-s − 0.204·24-s − 7/5·25-s − 0.392·26-s − 0.192·27-s + 0.371·29-s − 0.365·30-s + 1.07·31-s + 0.174·33-s − 0.857·34-s − 0.676·35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 736164 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 736164 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.342529185\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.342529185\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + T^{2} \) |
| 3 | $C_2$ | \( 1 - T + T^{2} \) |
| 11 | $C_2$ | \( 1 - T + T^{2} \) |
| 13 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 - 2 T - 3 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 5 T + 8 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 5 T + 6 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 8 T + 41 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 2 T - 25 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 2 T - 37 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 4 T - 27 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - T - 58 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 11 T + 60 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 2 T - 63 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 2 T - 85 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 19 T + p T^{2} ) \) |
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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
−10.54298248671324550038453604748, −9.896988220614733028413819635255, −9.534565387747137288961807639639, −8.810633868796006251179546082571, −8.749746893915327223239997630009, −8.382081225215226294899857127661, −7.86931391100553822103305836229, −7.24054507725785735858594045631, −7.20594586350111001523392886684, −6.34576195383975651338000875377, −6.21946148150173000014127390597, −5.19916201482602066574246735125, −5.09961086730724949522335620101, −4.41072817172552002562531423737, −4.10578347140989261502493534280, −3.70150179719989405192888694377, −2.99943053835457662284924266747, −2.35596969912902314816484243583, −1.91116599211117776189249755055, −0.63333450127094566975178159467,
0.63333450127094566975178159467, 1.91116599211117776189249755055, 2.35596969912902314816484243583, 2.99943053835457662284924266747, 3.70150179719989405192888694377, 4.10578347140989261502493534280, 4.41072817172552002562531423737, 5.09961086730724949522335620101, 5.19916201482602066574246735125, 6.21946148150173000014127390597, 6.34576195383975651338000875377, 7.20594586350111001523392886684, 7.24054507725785735858594045631, 7.86931391100553822103305836229, 8.382081225215226294899857127661, 8.749746893915327223239997630009, 8.810633868796006251179546082571, 9.534565387747137288961807639639, 9.896988220614733028413819635255, 10.54298248671324550038453604748
