| L(s) = 1 | + 11·5-s + 5·11-s − 5·13-s + 100·17-s − 67·19-s − 76·23-s − 111·25-s − 275·29-s − 362·31-s − 5·37-s − 162·41-s + 721·43-s − 216·47-s − 495·53-s + 55·55-s + 173·59-s + 532·61-s − 55·65-s + 111·67-s − 1.60e3·71-s − 1.21e3·73-s + 1.46e3·79-s − 1.40e3·83-s + 1.10e3·85-s − 1.97e3·89-s − 737·95-s − 561·97-s + ⋯ |
| L(s) = 1 | + 0.983·5-s + 0.137·11-s − 0.106·13-s + 1.42·17-s − 0.808·19-s − 0.689·23-s − 0.887·25-s − 1.76·29-s − 2.09·31-s − 0.0222·37-s − 0.617·41-s + 2.55·43-s − 0.670·47-s − 1.28·53-s + 0.134·55-s + 0.381·59-s + 1.11·61-s − 0.104·65-s + 0.202·67-s − 2.67·71-s − 1.94·73-s + 2.07·79-s − 1.86·83-s + 1.40·85-s − 2.35·89-s − 0.795·95-s − 0.587·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3111696 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3111696 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 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
−8.805448595766898552985203162745, −8.396493289856398931499217098088, −7.78371951264316977198834211651, −7.66784074350945642071579570999, −7.13452741454841205396870726304, −6.87097750727987642237460499925, −6.05001126702932798208359201871, −5.88796066711104091045766155620, −5.48874988266167824815658396576, −5.47133962186042415195748808409, −4.35308793206176351954497301295, −4.34649364899786603094737264772, −3.50876373457885020965654768839, −3.41420927824841122146531984273, −2.49850407740884715501161832479, −2.13333575779049972860808218319, −1.58798610577861464682137493355, −1.25376567857062498676501498342, 0, 0,
1.25376567857062498676501498342, 1.58798610577861464682137493355, 2.13333575779049972860808218319, 2.49850407740884715501161832479, 3.41420927824841122146531984273, 3.50876373457885020965654768839, 4.34649364899786603094737264772, 4.35308793206176351954497301295, 5.47133962186042415195748808409, 5.48874988266167824815658396576, 5.88796066711104091045766155620, 6.05001126702932798208359201871, 6.87097750727987642237460499925, 7.13452741454841205396870726304, 7.66784074350945642071579570999, 7.78371951264316977198834211651, 8.396493289856398931499217098088, 8.805448595766898552985203162745
