| L(s) = 1 | − 4·2-s + 12·4-s − 10·5-s + 27·7-s − 32·8-s + 40·10-s − 9·11-s − 26·13-s − 108·14-s + 80·16-s − 31·17-s + 96·19-s − 120·20-s + 36·22-s − 33·23-s + 75·25-s + 104·26-s + 324·28-s − 246·29-s − 34·31-s − 192·32-s + 124·34-s − 270·35-s + 105·37-s − 384·38-s + 320·40-s − 361·41-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 3/2·4-s − 0.894·5-s + 1.45·7-s − 1.41·8-s + 1.26·10-s − 0.246·11-s − 0.554·13-s − 2.06·14-s + 5/4·16-s − 0.442·17-s + 1.15·19-s − 1.34·20-s + 0.348·22-s − 0.299·23-s + 3/5·25-s + 0.784·26-s + 2.18·28-s − 1.57·29-s − 0.196·31-s − 1.06·32-s + 0.625·34-s − 1.30·35-s + 0.466·37-s − 1.63·38-s + 1.26·40-s − 1.37·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1368900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1368900 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 + p T )^{2} \) |
| 13 | $C_1$ | \( ( 1 + p T )^{2} \) |
| good | 7 | $D_{4}$ | \( 1 - 27 T + 348 T^{2} - 27 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 9 T + 2162 T^{2} + 9 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 31 T + 9546 T^{2} + 31 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 96 T + 7698 T^{2} - 96 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 33 T + 19924 T^{2} + 33 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 246 T + 61826 T^{2} + 246 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 34 T + 41142 T^{2} + 34 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 105 T + 16140 T^{2} - 105 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 361 T + 157416 T^{2} + 361 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 452 T + 201766 T^{2} - 452 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 382 T + 75566 T^{2} + 382 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 533 T + 343284 T^{2} + 533 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 340 T + 31782 T^{2} + 340 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 897 T + 650432 T^{2} + 897 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 684 T + 710166 T^{2} - 684 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 411 T + 745046 T^{2} - 411 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 78 T + 760826 T^{2} - 78 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 871 T + 1150246 T^{2} - 871 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 264 T + 628262 T^{2} + 264 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 323 T + 1110864 T^{2} + 323 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 3063 T + 4170318 T^{2} + 3063 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−9.163165057472544142746821493373, −8.821027905238353031133333634232, −8.193222152339351078830544889861, −7.995932425745842449630735537931, −7.59441043220530818657687585206, −7.59127925285651243371833658561, −6.75561734337582476100760453713, −6.68574355675560282509771117533, −5.68539376228185735085285964703, −5.55319297491802789499985498078, −4.73752549565999840846304590578, −4.65799530443363964645677886295, −3.76467358742085566149747280866, −3.39158321273411768841874542509, −2.64678085677747057096248587692, −2.15606590770506420611775594333, −1.46823731135779827907766881250, −1.14481181867308040799831149816, 0, 0,
1.14481181867308040799831149816, 1.46823731135779827907766881250, 2.15606590770506420611775594333, 2.64678085677747057096248587692, 3.39158321273411768841874542509, 3.76467358742085566149747280866, 4.65799530443363964645677886295, 4.73752549565999840846304590578, 5.55319297491802789499985498078, 5.68539376228185735085285964703, 6.68574355675560282509771117533, 6.75561734337582476100760453713, 7.59127925285651243371833658561, 7.59441043220530818657687585206, 7.995932425745842449630735537931, 8.193222152339351078830544889861, 8.821027905238353031133333634232, 9.163165057472544142746821493373
