| L(s) = 1 | − 6·7-s − 2·11-s − 36·23-s − 10·25-s − 54·29-s − 12·37-s + 40·43-s − 6·49-s + 160·53-s + 240·67-s + 40·71-s + 12·77-s − 42·79-s − 24·107-s − 162·109-s + 4·113-s − 189·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 216·161-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | − 6/7·7-s − 0.181·11-s − 1.56·23-s − 2/5·25-s − 1.86·29-s − 0.324·37-s + 0.930·43-s − 0.122·49-s + 3.01·53-s + 3.58·67-s + 0.563·71-s + 0.155·77-s − 0.531·79-s − 0.224·107-s − 1.48·109-s + 0.0353·113-s − 1.56·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 1.34·161-s + 0.00613·163-s + 0.00598·167-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.1986100227\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1986100227\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 + 6 T + 6 p T^{2} + 6 p^{2} T^{3} + p^{4} T^{4} \) |
| good | 11 | $D_{4}$ | \( ( 1 + T + 96 T^{2} + p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 - 667 T^{2} + 168328 T^{4} - 667 p^{4} T^{6} + p^{8} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 791 T^{2} + 290556 T^{4} - 791 p^{4} T^{6} + p^{8} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 - 1180 T^{2} + 592102 T^{4} - 1180 p^{4} T^{6} + p^{8} T^{8} \) |
| 23 | $D_{4}$ | \( ( 1 + 18 T + 1074 T^{2} + 18 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 29 | $D_{4}$ | \( ( 1 + 27 T + 1068 T^{2} + 27 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 - 880 T^{2} - 112418 T^{4} - 880 p^{4} T^{6} + p^{8} T^{8} \) |
| 37 | $D_{4}$ | \( ( 1 + 6 T - 438 T^{2} + 6 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 41 | $D_4\times C_2$ | \( 1 - 3328 T^{2} + 7633918 T^{4} - 3328 p^{4} T^{6} + p^{8} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 - 20 T + 2758 T^{2} - 20 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 - 2335 T^{2} + 2439052 T^{4} - 2335 p^{4} T^{6} + p^{8} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 - 80 T + 6958 T^{2} - 80 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 - 3260 T^{2} + 11915622 T^{4} - 3260 p^{4} T^{6} + p^{8} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 + 1760 T^{2} - 11282978 T^{4} + 1760 p^{4} T^{6} + p^{8} T^{8} \) |
| 67 | $D_{4}$ | \( ( 1 - 120 T + 10238 T^{2} - 120 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 71 | $D_{4}$ | \( ( 1 - 20 T + 6022 T^{2} - 20 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 4272 T^{2} + 57425438 T^{4} - 4272 p^{4} T^{6} + p^{8} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 + 21 T + 12576 T^{2} + 21 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 14140 T^{2} + 99906982 T^{4} - 14140 p^{4} T^{6} + p^{8} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 + 5900 T^{2} + 132685222 T^{4} + 5900 p^{4} T^{6} + p^{8} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 - 33447 T^{2} + 452943068 T^{4} - 33447 p^{4} T^{6} + p^{8} T^{8} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.73937222535700455592991463689, −6.55942116709494506025444919669, −6.32717640829836184098916510611, −6.08999994458551311150559696687, −5.73024065987792912245842072677, −5.65330225065935670923876003890, −5.42977432111391350320214640594, −5.18451024902275630932191042849, −5.04017364224410665190020618517, −4.84753671835970109383060800713, −4.16308393989939378357457540207, −4.06898611476118634606572744428, −3.86349522800417134455730241083, −3.82598917757246194257029541609, −3.71856612698442719855073887400, −3.18836003126865036353391583665, −2.70432894222050300833742856850, −2.69992610816145536718361233857, −2.40584984900563348741418086853, −2.01958241435933797844219318392, −1.87136608421049821245770362059, −1.37130403187413713846809430517, −1.00445839125311702250074654177, −0.57957659018057061805430203739, −0.07724832312289029001202236298,
0.07724832312289029001202236298, 0.57957659018057061805430203739, 1.00445839125311702250074654177, 1.37130403187413713846809430517, 1.87136608421049821245770362059, 2.01958241435933797844219318392, 2.40584984900563348741418086853, 2.69992610816145536718361233857, 2.70432894222050300833742856850, 3.18836003126865036353391583665, 3.71856612698442719855073887400, 3.82598917757246194257029541609, 3.86349522800417134455730241083, 4.06898611476118634606572744428, 4.16308393989939378357457540207, 4.84753671835970109383060800713, 5.04017364224410665190020618517, 5.18451024902275630932191042849, 5.42977432111391350320214640594, 5.65330225065935670923876003890, 5.73024065987792912245842072677, 6.08999994458551311150559696687, 6.32717640829836184098916510611, 6.55942116709494506025444919669, 6.73937222535700455592991463689
