| L(s) = 1 | − 4·2-s + 2·3-s + 12·4-s + 12·5-s − 8·6-s − 18·7-s − 32·8-s − 24·9-s − 48·10-s + 66·11-s + 24·12-s − 10·13-s + 72·14-s + 24·15-s + 80·16-s − 168·17-s + 96·18-s + 42·19-s + 144·20-s − 36·21-s − 264·22-s − 64·24-s − 139·25-s + 40·26-s − 50·27-s − 216·28-s + 210·29-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 0.384·3-s + 3/2·4-s + 1.07·5-s − 0.544·6-s − 0.971·7-s − 1.41·8-s − 8/9·9-s − 1.51·10-s + 1.80·11-s + 0.577·12-s − 0.213·13-s + 1.37·14-s + 0.413·15-s + 5/4·16-s − 2.39·17-s + 1.25·18-s + 0.507·19-s + 1.60·20-s − 0.374·21-s − 2.55·22-s − 0.544·24-s − 1.11·25-s + 0.301·26-s − 0.356·27-s − 1.45·28-s + 1.34·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1119364 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1119364 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_1$ | \( ( 1 + p T )^{2} \) |
| 23 | | \( 1 \) |
| good | 3 | $D_{4}$ | \( 1 - 2 T + 28 T^{2} - 2 p^{3} T^{3} + p^{6} T^{4} \) |
| 5 | $D_{4}$ | \( 1 - 12 T + 283 T^{2} - 12 p^{3} T^{3} + p^{6} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 18 T + 620 T^{2} + 18 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 6 p T + 3508 T^{2} - 6 p^{4} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 10 T + 3447 T^{2} + 10 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 168 T + 15910 T^{2} + 168 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 42 T + 14156 T^{2} - 42 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 210 T + 38635 T^{2} - 210 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 32 T + 54546 T^{2} - 32 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 396 T + 78302 T^{2} + 396 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 30 T + 47239 T^{2} + 30 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 696 T + 279818 T^{2} - 696 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 78 T + 176092 T^{2} + 78 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 720 T + 371047 T^{2} + 720 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 774 T + 552724 T^{2} + 774 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 528 T + 303335 T^{2} + 528 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 924 T + 778670 T^{2} - 924 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 786 T + 318148 T^{2} + 786 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 422 T + 393903 T^{2} - 422 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 204 T + 764630 T^{2} - 204 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 888 T + 338458 T^{2} + 888 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 132 T + 1153219 T^{2} - 132 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 192 T + 1057319 T^{2} - 192 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−9.250043367625321545615906678274, −9.092742401190132900808871105162, −8.594893788086420899197467070890, −8.457677152547828288866566430439, −7.55349646059217622891521771735, −7.42477455652158998222883807292, −6.60553467707085990641240669625, −6.45570712546408860252615370934, −6.16457718947574778830211675496, −5.91678897802811324389482476920, −4.96705896118526544403344977278, −4.52977895827483529936413968244, −3.70402594518788603323360969936, −3.36023048783361778772991302143, −2.56515887749423674454264965355, −2.35714191811717670655996468153, −1.64064834414907763046613518690, −1.20065608563026973588217346778, 0, 0,
1.20065608563026973588217346778, 1.64064834414907763046613518690, 2.35714191811717670655996468153, 2.56515887749423674454264965355, 3.36023048783361778772991302143, 3.70402594518788603323360969936, 4.52977895827483529936413968244, 4.96705896118526544403344977278, 5.91678897802811324389482476920, 6.16457718947574778830211675496, 6.45570712546408860252615370934, 6.60553467707085990641240669625, 7.42477455652158998222883807292, 7.55349646059217622891521771735, 8.457677152547828288866566430439, 8.594893788086420899197467070890, 9.092742401190132900808871105162, 9.250043367625321545615906678274
