| L(s) = 1 | − 2·3-s + 10·5-s − 51·9-s + 26·11-s + 30·13-s − 20·15-s − 78·17-s + 112·19-s − 36·23-s + 75·25-s + 158·27-s − 66·29-s − 112·31-s − 52·33-s − 140·37-s − 60·39-s + 280·41-s − 40·43-s − 510·45-s + 366·47-s + 156·51-s + 164·53-s + 260·55-s − 224·57-s + 76·59-s − 932·61-s + 300·65-s + ⋯ |
| L(s) = 1 | − 0.384·3-s + 0.894·5-s − 1.88·9-s + 0.712·11-s + 0.640·13-s − 0.344·15-s − 1.11·17-s + 1.35·19-s − 0.326·23-s + 3/5·25-s + 1.12·27-s − 0.422·29-s − 0.648·31-s − 0.274·33-s − 0.622·37-s − 0.246·39-s + 1.06·41-s − 0.141·43-s − 1.68·45-s + 1.13·47-s + 0.428·51-s + 0.425·53-s + 0.637·55-s − 0.520·57-s + 0.167·59-s − 1.95·61-s + 0.572·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3841600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3841600 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 - p T )^{2} \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2$ | \( ( 1 + T + p^{3} T^{2} )^{2} \) |
| 11 | $D_{4}$ | \( 1 - 26 T + 155 T^{2} - 26 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 30 T + 1943 T^{2} - 30 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 78 T + 8671 T^{2} + 78 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 112 T + 14178 T^{2} - 112 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 36 T + 21982 T^{2} + 36 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 66 T + 39163 T^{2} + 66 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 112 T + 38634 T^{2} + 112 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 140 T + 103530 T^{2} + 140 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 280 T + 133358 T^{2} - 280 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 40 T + 135330 T^{2} + 40 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 366 T + 198319 T^{2} - 366 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 164 T + 208142 T^{2} - 164 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 76 T + 401498 T^{2} - 76 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 932 T + 604218 T^{2} + 932 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 196 T + 568314 T^{2} + 196 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 240 T + 558958 T^{2} + 240 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 132 T - 84634 T^{2} + 132 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 946 T + 757563 T^{2} + 946 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 704 T + 999878 T^{2} + 704 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 2020 T + 2298914 T^{2} + 2020 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 322 T + 1784367 T^{2} - 322 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−8.724482178090384378082863584275, −8.444208734277739520851898619428, −7.82406753951102624124972601276, −7.47814346347084238007213353086, −6.84221468737113752496065836147, −6.70006921895094687220994151175, −6.00960657438519722730149861335, −5.88007296607527897865172573111, −5.47048046903436683709707826537, −5.32720601185331763819246455963, −4.37255517155367125421985448894, −4.34212879859241479321917492529, −3.38515666539870324454905834446, −3.25431998006473711681285086600, −2.46718444574408537922907640995, −2.33739024406043314657905569946, −1.28969456316652274269026846516, −1.22795038148844226550472145070, 0, 0,
1.22795038148844226550472145070, 1.28969456316652274269026846516, 2.33739024406043314657905569946, 2.46718444574408537922907640995, 3.25431998006473711681285086600, 3.38515666539870324454905834446, 4.34212879859241479321917492529, 4.37255517155367125421985448894, 5.32720601185331763819246455963, 5.47048046903436683709707826537, 5.88007296607527897865172573111, 6.00960657438519722730149861335, 6.70006921895094687220994151175, 6.84221468737113752496065836147, 7.47814346347084238007213353086, 7.82406753951102624124972601276, 8.444208734277739520851898619428, 8.724482178090384378082863584275
