| L(s) = 1 | − 4·2-s + 6·3-s + 12·4-s + 7·5-s − 24·6-s − 14·7-s − 32·8-s + 27·9-s − 28·10-s − 22·11-s + 72·12-s − 27·13-s + 56·14-s + 42·15-s + 80·16-s − 64·17-s − 108·18-s − 117·19-s + 84·20-s − 84·21-s + 88·22-s − 60·23-s − 192·24-s − 159·25-s + 108·26-s + 108·27-s − 168·28-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 1.15·3-s + 3/2·4-s + 0.626·5-s − 1.63·6-s − 0.755·7-s − 1.41·8-s + 9-s − 0.885·10-s − 0.603·11-s + 1.73·12-s − 0.576·13-s + 1.06·14-s + 0.722·15-s + 5/4·16-s − 0.913·17-s − 1.41·18-s − 1.41·19-s + 0.939·20-s − 0.872·21-s + 0.852·22-s − 0.543·23-s − 1.63·24-s − 1.27·25-s + 0.814·26-s + 0.769·27-s − 1.13·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 213444 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 213444 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 + p T )^{2} \) |
| 3 | $C_1$ | \( ( 1 - p T )^{2} \) |
| 7 | $C_1$ | \( ( 1 + p T )^{2} \) |
| 11 | $C_1$ | \( ( 1 + p T )^{2} \) |
| good | 5 | $D_{4}$ | \( 1 - 7 T + 208 T^{2} - 7 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 27 T + 3220 T^{2} + 27 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 64 T + 3038 T^{2} + 64 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 117 T + 12746 T^{2} + 117 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 60 T + 3534 T^{2} + 60 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 135 T + 50676 T^{2} + 135 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 134 T + 63854 T^{2} + 134 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 257 T + 83912 T^{2} + 257 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 86 T + 61354 T^{2} - 86 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 208 T + 138582 T^{2} + 208 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 227 T + 216134 T^{2} - 227 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 378 T + 331522 T^{2} - 378 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 371 T + 314914 T^{2} - 371 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 872 T + 643190 T^{2} + 872 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 269 T + 390410 T^{2} - 269 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 100 T - 11666 T^{2} + 100 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 1493 T + 1328732 T^{2} + 1493 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 1260 T + 1187678 T^{2} + 1260 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 662 T + 1247710 T^{2} + 662 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 80 T + 1306510 T^{2} - 80 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 1666 T + 2501658 T^{2} + 1666 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−10.16933865080689214203769583060, −9.903219324633934792534631902231, −9.431475515370714926654991167079, −9.130096012767526933009061710563, −8.534250140143252152757838634991, −8.482446599626273860614338422963, −7.59586603242887826017838926959, −7.52194347098371179660686532108, −6.73711294715241412479607572727, −6.61233875451635945241224314355, −5.63422774288265268935172497880, −5.57095216510323526417887390125, −4.22335326160152108551956636635, −4.02947376000296621764478034450, −3.00591887597230512806242077772, −2.61051501755686523748141273232, −1.93560925984882807555525163335, −1.65857818070270074300461918472, 0, 0,
1.65857818070270074300461918472, 1.93560925984882807555525163335, 2.61051501755686523748141273232, 3.00591887597230512806242077772, 4.02947376000296621764478034450, 4.22335326160152108551956636635, 5.57095216510323526417887390125, 5.63422774288265268935172497880, 6.61233875451635945241224314355, 6.73711294715241412479607572727, 7.52194347098371179660686532108, 7.59586603242887826017838926959, 8.482446599626273860614338422963, 8.534250140143252152757838634991, 9.130096012767526933009061710563, 9.431475515370714926654991167079, 9.903219324633934792534631902231, 10.16933865080689214203769583060
