| L(s) = 1 | − 2·2-s − 13·4-s + 44·8-s − 116·13-s + 101·16-s + 34·17-s + 160·19-s − 25·25-s + 232·26-s − 614·32-s − 68·34-s − 320·38-s + 544·43-s + 928·47-s + 490·49-s + 50·50-s + 1.50e3·52-s − 1.28e3·53-s + 360·59-s − 361·64-s − 1.84e3·67-s − 442·68-s − 2.08e3·76-s − 1.10e3·83-s − 1.08e3·86-s − 2.98e3·89-s − 1.85e3·94-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.62·4-s + 1.94·8-s − 2.47·13-s + 1.57·16-s + 0.485·17-s + 1.93·19-s − 1/5·25-s + 1.74·26-s − 3.39·32-s − 0.342·34-s − 1.36·38-s + 1.92·43-s + 2.88·47-s + 10/7·49-s + 0.141·50-s + 4.02·52-s − 3.32·53-s + 0.794·59-s − 0.705·64-s − 3.36·67-s − 0.788·68-s − 3.13·76-s − 1.45·83-s − 1.36·86-s − 3.54·89-s − 2.03·94-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 585225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 585225 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.2879729961\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2879729961\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 5 | $C_2$ | \( 1 + p^{2} T^{2} \) |
| 17 | $C_2$ | \( 1 - 2 p T + p^{3} T^{2} \) |
| good | 2 | $C_2$ | \( ( 1 + T + p^{3} T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 - 10 p^{2} T^{2} + p^{6} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 2262 T^{2} + p^{6} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 58 T + p^{3} T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 80 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 10410 T^{2} + p^{6} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 32902 T^{2} + p^{6} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 54682 T^{2} + p^{6} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 83350 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 127842 T^{2} + p^{6} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 272 T + p^{3} T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 464 T + p^{3} T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 642 T + p^{3} T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 180 T + p^{3} T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 - 441862 T^{2} + p^{6} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 924 T + p^{3} T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 707722 T^{2} + p^{6} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 92450 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 793478 T^{2} + p^{6} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 552 T + p^{3} T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 1490 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 68030 T^{2} + p^{6} T^{4} \) |
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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
−10.10264783119262187619488115721, −9.571977700571254408161654879961, −9.495609738217990034205910716316, −8.844084924294353468627204017788, −8.739723734303078658696717873776, −7.78557468793819287209879099751, −7.73611160198586805057705096925, −7.22448373063385657719412863422, −7.21321938599776353510892862110, −5.81531775741286154918724597497, −5.73251330401217675200044233329, −5.10003192353599843728172474124, −4.83218900968117299092467381835, −4.12399991486893525580221325055, −4.02294700994622702641276324468, −2.82455133886230251918919275996, −2.78905387016857489907687681912, −1.50241701518432985227705302432, −1.02728935031978317871780645521, −0.20962525888679202069555978487,
0.20962525888679202069555978487, 1.02728935031978317871780645521, 1.50241701518432985227705302432, 2.78905387016857489907687681912, 2.82455133886230251918919275996, 4.02294700994622702641276324468, 4.12399991486893525580221325055, 4.83218900968117299092467381835, 5.10003192353599843728172474124, 5.73251330401217675200044233329, 5.81531775741286154918724597497, 7.21321938599776353510892862110, 7.22448373063385657719412863422, 7.73611160198586805057705096925, 7.78557468793819287209879099751, 8.739723734303078658696717873776, 8.844084924294353468627204017788, 9.495609738217990034205910716316, 9.571977700571254408161654879961, 10.10264783119262187619488115721
