| L(s) = 1 | − 6·3-s + 11·5-s + 27·9-s + 5·11-s + 5·13-s − 66·15-s + 100·17-s − 67·19-s − 76·23-s − 111·25-s − 108·27-s + 275·29-s − 362·31-s − 30·33-s − 5·37-s − 30·39-s − 162·41-s − 721·43-s + 297·45-s + 216·47-s − 600·51-s + 495·53-s + 55·55-s + 402·57-s − 173·59-s − 532·61-s + 55·65-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 0.983·5-s + 9-s + 0.137·11-s + 0.106·13-s − 1.13·15-s + 1.42·17-s − 0.808·19-s − 0.689·23-s − 0.887·25-s − 0.769·27-s + 1.76·29-s − 2.09·31-s − 0.158·33-s − 0.0222·37-s − 0.123·39-s − 0.617·41-s − 2.55·43-s + 0.983·45-s + 0.670·47-s − 1.64·51-s + 1.28·53-s + 0.134·55-s + 0.934·57-s − 0.381·59-s − 1.11·61-s + 0.104·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5531904 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5531904 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 + p T )^{2} \) |
| 7 | | \( 1 \) |
| good | 5 | $D_{4}$ | \( 1 - 11 T + 232 T^{2} - 11 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 5 T + 304 T^{2} - 5 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 5 T + 3194 T^{2} - 5 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 100 T + 11554 T^{2} - 100 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 67 T + 14406 T^{2} + 67 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 76 T + 6478 T^{2} + 76 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 275 T + 61846 T^{2} - 275 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 362 T + 73043 T^{2} + 362 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 5 T + 3606 T^{2} + 5 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 162 T + 128770 T^{2} + 162 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 721 T + 288540 T^{2} + 721 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 216 T + 156778 T^{2} - 216 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 495 T + 355102 T^{2} - 495 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 173 T + 282706 T^{2} + 173 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 532 T + 447518 T^{2} + 532 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 111 T + 318532 T^{2} + 111 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 1600 T + 1262410 T^{2} + 1600 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 1215 T + 1011556 T^{2} - 1215 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 1460 T + 1333505 T^{2} + 1460 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 1409 T + 1147696 T^{2} - 1409 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 1974 T + 2298994 T^{2} + 1974 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 561 T + 1863448 T^{2} - 561 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−8.489791454138755866547818134667, −7.968205162927056310101032043457, −7.62950038853795787514120085619, −7.25162513431426703737584226324, −6.68849113989619723462544029894, −6.47750141423672492360192101737, −5.98140596686762015832861944777, −5.77833262545322139391729589885, −5.31132178128514068598955549039, −5.13498799704313310110407999863, −4.41523652175255578838201776441, −4.17235669819090669661984352588, −3.47117420721504710550215749721, −3.21843617269214504560827855104, −2.37485102426504338547108176065, −1.89476993831452844571421132958, −1.47763718808514126962142306682, −1.03897870946516744300063282953, 0, 0,
1.03897870946516744300063282953, 1.47763718808514126962142306682, 1.89476993831452844571421132958, 2.37485102426504338547108176065, 3.21843617269214504560827855104, 3.47117420721504710550215749721, 4.17235669819090669661984352588, 4.41523652175255578838201776441, 5.13498799704313310110407999863, 5.31132178128514068598955549039, 5.77833262545322139391729589885, 5.98140596686762015832861944777, 6.47750141423672492360192101737, 6.68849113989619723462544029894, 7.25162513431426703737584226324, 7.62950038853795787514120085619, 7.968205162927056310101032043457, 8.489791454138755866547818134667
