| L(s) = 1 | + 2·3-s − 3·4-s − 4·5-s − 7-s + 9-s − 6·12-s − 8·15-s + 5·16-s + 8·17-s + 12·20-s − 2·21-s + 2·25-s − 4·27-s + 3·28-s + 4·35-s − 3·36-s − 12·37-s + 8·41-s + 24·43-s − 4·45-s − 20·47-s + 10·48-s + 49-s + 16·51-s + 4·59-s + 24·60-s − 63-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 3/2·4-s − 1.78·5-s − 0.377·7-s + 1/3·9-s − 1.73·12-s − 2.06·15-s + 5/4·16-s + 1.94·17-s + 2.68·20-s − 0.436·21-s + 2/5·25-s − 0.769·27-s + 0.566·28-s + 0.676·35-s − 1/2·36-s − 1.97·37-s + 1.24·41-s + 3.65·43-s − 0.596·45-s − 2.91·47-s + 1.44·48-s + 1/7·49-s + 2.24·51-s + 0.520·59-s + 3.09·60-s − 0.125·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 373527 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 373527 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| 7 | $C_1$ | \( 1 + T \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 2 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 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
−8.503382678912186991908631693316, −7.959279630374951452453810048684, −7.82867506007281568614416512666, −7.42272354779304253907064109454, −6.84576801400653059993297260469, −5.96443978110022033983848919969, −5.47259073254467597109793181667, −5.02042250101516742575983822668, −4.11624652535699675148140948571, −4.07930883034575373640745113861, −3.42834085160890094357341585879, −3.26888344602981049454952325554, −2.30344485967966511448007334997, −1.01039812063869528093138870562, 0,
1.01039812063869528093138870562, 2.30344485967966511448007334997, 3.26888344602981049454952325554, 3.42834085160890094357341585879, 4.07930883034575373640745113861, 4.11624652535699675148140948571, 5.02042250101516742575983822668, 5.47259073254467597109793181667, 5.96443978110022033983848919969, 6.84576801400653059993297260469, 7.42272354779304253907064109454, 7.82867506007281568614416512666, 7.959279630374951452453810048684, 8.503382678912186991908631693316
