| L(s) = 1 | − 8·2-s + 312·5-s − 323·7-s + 512·8-s − 2.49e3·10-s + 3.72e3·11-s + 1.41e4·13-s + 2.58e3·14-s − 4.09e3·16-s − 3.18e4·17-s + 4.48e4·19-s − 2.97e4·22-s + 5.77e4·23-s + 7.81e4·25-s − 1.13e5·26-s + 1.66e5·29-s − 9.48e4·31-s + 2.54e5·34-s − 1.00e5·35-s + 9.07e5·37-s − 3.58e5·38-s + 1.59e5·40-s + 6.27e5·41-s + 4.24e4·43-s − 4.62e5·46-s − 1.23e6·47-s + 8.23e5·49-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.11·5-s − 0.355·7-s + 0.353·8-s − 0.789·10-s + 0.842·11-s + 1.78·13-s + 0.251·14-s − 1/4·16-s − 1.57·17-s + 1.49·19-s − 0.595·22-s + 0.990·23-s + 25-s − 1.26·26-s + 1.26·29-s − 0.571·31-s + 1.11·34-s − 0.397·35-s + 2.94·37-s − 1.06·38-s + 0.394·40-s + 1.42·41-s + 0.0814·43-s − 0.700·46-s − 1.73·47-s + 49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 26244 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 26244 ^{s/2} \, \Gamma_{\C}(s+7/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(3.111840590\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.111840590\) |
| \(L(\frac{9}{2})\) |
|
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 | $C_2$ | \( 1 + p^{3} T + p^{6} T^{2} \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 - 312 T + 19219 T^{2} - 312 p^{7} T^{3} + p^{14} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 323 T - 719214 T^{2} + 323 p^{7} T^{3} + p^{14} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 3720 T - 5648771 T^{2} - 3720 p^{7} T^{3} + p^{14} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 14179 T + 138295524 T^{2} - 14179 p^{7} T^{3} + p^{14} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 936 p T + p^{7} T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 22421 T + p^{7} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 57768 T - 67683623 T^{2} - 57768 p^{7} T^{3} + p^{14} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 166656 T + 10524346027 T^{2} - 166656 p^{7} T^{3} + p^{14} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 94820 T - 18521781711 T^{2} + 94820 p^{7} T^{3} + p^{14} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 453971 T + p^{7} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 627072 T + 198465019303 T^{2} - 627072 p^{7} T^{3} + p^{14} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 42472 T - 270014740323 T^{2} - 42472 p^{7} T^{3} + p^{14} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 1235256 T + 1019234265073 T^{2} + 1235256 p^{7} T^{3} + p^{14} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 107280 T + p^{7} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 2479224 T + 3657900157357 T^{2} + 2479224 p^{7} T^{3} + p^{14} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 2874383 T + 5119334794668 T^{2} + 2874383 p^{7} T^{3} + p^{14} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 1501097 T - 3807419401914 T^{2} + 1501097 p^{7} T^{3} + p^{14} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 4733136 T + p^{7} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 85111 T + p^{7} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 1180819 T - 17809575475398 T^{2} - 1180819 p^{7} T^{3} + p^{14} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 1116528 T - 25889416214843 T^{2} + 1116528 p^{7} T^{3} + p^{14} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 9368136 T + p^{7} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 2039995 T - 76636704878088 T^{2} - 2039995 p^{7} T^{3} + p^{14} T^{4} \) |
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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
−11.54982270745521962193746299441, −11.00984574633140183309357516436, −10.98542177571393625142763392530, −10.16641145952829371959858174007, −9.602918432552615002593038358053, −9.138972003382514074366718933482, −9.050731658935884637861966736338, −8.410257668707143090069459493226, −7.69969791124155265410332606926, −7.05972311863445637643009704978, −6.39884913741107267959060441681, −6.11294517695656758348573004458, −5.58249743083386214584894121231, −4.45548341869070580586840537060, −4.32006346023210792978053981346, −3.04798904711665549597888295593, −2.79753116843316276213976782823, −1.50676143875448754014717157756, −1.31333493666739910456883580042, −0.58559212633771622085649779422,
0.58559212633771622085649779422, 1.31333493666739910456883580042, 1.50676143875448754014717157756, 2.79753116843316276213976782823, 3.04798904711665549597888295593, 4.32006346023210792978053981346, 4.45548341869070580586840537060, 5.58249743083386214584894121231, 6.11294517695656758348573004458, 6.39884913741107267959060441681, 7.05972311863445637643009704978, 7.69969791124155265410332606926, 8.410257668707143090069459493226, 9.050731658935884637861966736338, 9.138972003382514074366718933482, 9.602918432552615002593038358053, 10.16641145952829371959858174007, 10.98542177571393625142763392530, 11.00984574633140183309357516436, 11.54982270745521962193746299441
