| L(s) = 1 | − 2·2-s + 6·5-s + 16·7-s + 8·8-s − 12·10-s + 12·11-s − 38·13-s − 32·14-s − 16·16-s + 252·17-s + 40·19-s − 24·22-s + 168·23-s + 125·25-s + 76·26-s + 30·29-s + 88·31-s − 504·34-s + 96·35-s + 508·37-s − 80·38-s + 48·40-s + 42·41-s + 52·43-s − 336·46-s − 96·47-s + 343·49-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.536·5-s + 0.863·7-s + 0.353·8-s − 0.379·10-s + 0.328·11-s − 0.810·13-s − 0.610·14-s − 1/4·16-s + 3.59·17-s + 0.482·19-s − 0.232·22-s + 1.52·23-s + 25-s + 0.573·26-s + 0.192·29-s + 0.509·31-s − 2.54·34-s + 0.463·35-s + 2.25·37-s − 0.341·38-s + 0.189·40-s + 0.159·41-s + 0.184·43-s − 1.07·46-s − 0.297·47-s + 49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 26244 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 26244 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.393910805\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.393910805\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2$ | \( 1 + p T + p^{2} T^{2} \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 - 6 T - 89 T^{2} - 6 p^{3} T^{3} + p^{6} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 16 T - 87 T^{2} - 16 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 12 T - 1187 T^{2} - 12 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 38 T - 753 T^{2} + 38 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 126 T + p^{3} T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 20 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 168 T + 16057 T^{2} - 168 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 30 T - 23489 T^{2} - 30 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 88 T - 22047 T^{2} - 88 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 254 T + p^{3} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 42 T - 67157 T^{2} - 42 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 52 T - 76803 T^{2} - 52 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 96 T - 94607 T^{2} + 96 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 198 T + p^{3} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 660 T + 230221 T^{2} + 660 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 538 T + 62463 T^{2} - 538 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 884 T + 480693 T^{2} + 884 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 792 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 218 T + p^{3} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 520 T - 222639 T^{2} - 520 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 492 T - 329723 T^{2} + 492 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 810 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 1154 T + 419043 T^{2} + 1154 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−12.52672256973751990783629538386, −12.20839397221509747726704389949, −11.51037612533864586149893724782, −11.22510497842326500104040318332, −10.28851763362464824538447798509, −10.22591675239314802363510881460, −9.499692523440939874521279418580, −9.324928372625985479939020076123, −8.489242657620828730449982424819, −8.002542318996537816098215346162, −7.39304984983297156048979627129, −7.28963822072086638822669320879, −6.09384407451649237276719747587, −5.65329438305695635211099530227, −4.96916493000047226129319386859, −4.50686071330053995325001594584, −3.25969045505317201722044621140, −2.75760071290381303872345961737, −1.27481194623403663145947945777, −1.06071706954334300248305886668,
1.06071706954334300248305886668, 1.27481194623403663145947945777, 2.75760071290381303872345961737, 3.25969045505317201722044621140, 4.50686071330053995325001594584, 4.96916493000047226129319386859, 5.65329438305695635211099530227, 6.09384407451649237276719747587, 7.28963822072086638822669320879, 7.39304984983297156048979627129, 8.002542318996537816098215346162, 8.489242657620828730449982424819, 9.324928372625985479939020076123, 9.499692523440939874521279418580, 10.22591675239314802363510881460, 10.28851763362464824538447798509, 11.22510497842326500104040318332, 11.51037612533864586149893724782, 12.20839397221509747726704389949, 12.52672256973751990783629538386
