| L(s) = 1 | − 2-s − 47·4-s + 50·5-s − 98·7-s + 63·8-s − 50·10-s + 601·11-s − 577·13-s + 98·14-s + 1.20e3·16-s − 41·17-s + 630·19-s − 2.35e3·20-s − 601·22-s + 442·23-s + 1.87e3·25-s + 577·26-s + 4.60e3·28-s − 5.88e3·29-s − 396·31-s − 1.69e3·32-s + 41·34-s − 4.90e3·35-s − 8.90e3·37-s − 630·38-s + 3.15e3·40-s − 1.77e3·41-s + ⋯ |
| L(s) = 1 | − 0.176·2-s − 1.46·4-s + 0.894·5-s − 0.755·7-s + 0.348·8-s − 0.158·10-s + 1.49·11-s − 0.946·13-s + 0.133·14-s + 1.17·16-s − 0.0344·17-s + 0.400·19-s − 1.31·20-s − 0.264·22-s + 0.174·23-s + 3/5·25-s + 0.167·26-s + 1.11·28-s − 1.29·29-s − 0.0740·31-s − 0.292·32-s + 0.00608·34-s − 0.676·35-s − 1.06·37-s − 0.0707·38-s + 0.311·40-s − 0.164·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 99225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99225 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 - p^{2} T )^{2} \) |
| 7 | $C_1$ | \( ( 1 + p^{2} T )^{2} \) |
| good | 2 | $D_{4}$ | \( 1 + T + 3 p^{4} T^{2} + p^{5} T^{3} + p^{10} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 601 T + 259506 T^{2} - 601 p^{5} T^{3} + p^{10} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 577 T + 780172 T^{2} + 577 p^{5} T^{3} + p^{10} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 41 T + 1816368 T^{2} + 41 p^{5} T^{3} + p^{10} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 630 T + 4931238 T^{2} - 630 p^{5} T^{3} + p^{10} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 442 T - 299538 T^{2} - 442 p^{5} T^{3} + p^{10} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 5885 T + 35168948 T^{2} + 5885 p^{5} T^{3} + p^{10} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 396 T + 43423646 T^{2} + 396 p^{5} T^{3} + p^{10} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 8904 T + 132709718 T^{2} + 8904 p^{5} T^{3} + p^{10} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 1774 T - 4379094 T^{2} + 1774 p^{5} T^{3} + p^{10} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 27122 T + 464103742 T^{2} + 27122 p^{5} T^{3} + p^{10} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 21289 T + 531685238 T^{2} - 21289 p^{5} T^{3} + p^{10} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 55582 T + 1605381282 T^{2} - 55582 p^{5} T^{3} + p^{10} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 59600 T + 1881188438 T^{2} + 59600 p^{5} T^{3} + p^{10} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 51846 T + 1832119946 T^{2} + 51846 p^{5} T^{3} + p^{10} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 45344 T + 2875187158 T^{2} + 45344 p^{5} T^{3} + p^{10} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 80744 T + 4781205326 T^{2} + 80744 p^{5} T^{3} + p^{10} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 13532 T + 3906362902 T^{2} + 13532 p^{5} T^{3} + p^{10} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 51795 T + 1771153398 T^{2} + 51795 p^{5} T^{3} + p^{10} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 109828 T + 10643414822 T^{2} + 109828 p^{5} T^{3} + p^{10} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 37650 T + 10990453658 T^{2} - 37650 p^{5} T^{3} + p^{10} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 96339 T + 16335863448 T^{2} + 96339 p^{5} T^{3} + p^{10} T^{4} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.30070572094034656581720707076, −10.04913785178193353363361755884, −9.520440612253609428201484253248, −9.229467752114237855640760901308, −8.792427168583566349439060444848, −8.677638801117669153065151842697, −7.58311375807142811795467682041, −7.28979031650323968124547964654, −6.64723424087362840971323701027, −6.16849123800573892621289867332, −5.36575485034628472963006233604, −5.32597050109335014363708857242, −4.28836160925783180451674140082, −4.13135456458158458090289048099, −3.29832405754371658664629042988, −2.72616703758039750280551055310, −1.61463199393031977078664904262, −1.27176536806453243952899125631, 0, 0,
1.27176536806453243952899125631, 1.61463199393031977078664904262, 2.72616703758039750280551055310, 3.29832405754371658664629042988, 4.13135456458158458090289048099, 4.28836160925783180451674140082, 5.32597050109335014363708857242, 5.36575485034628472963006233604, 6.16849123800573892621289867332, 6.64723424087362840971323701027, 7.28979031650323968124547964654, 7.58311375807142811795467682041, 8.677638801117669153065151842697, 8.792427168583566349439060444848, 9.229467752114237855640760901308, 9.520440612253609428201484253248, 10.04913785178193353363361755884, 10.30070572094034656581720707076
