| L(s) = 1 | + 20·2-s − 20·3-s + 120·4-s + 250·5-s − 400·6-s − 640·8-s + 790·9-s + 5.00e3·10-s + 4.54e3·11-s − 2.40e3·12-s − 3.54e3·13-s − 5.00e3·15-s − 1.15e4·16-s + 2.73e4·17-s + 1.58e4·18-s − 3.87e4·19-s + 3.00e4·20-s + 9.08e4·22-s − 1.24e5·23-s + 1.28e4·24-s + 4.68e4·25-s − 7.08e4·26-s − 6.73e4·27-s − 7.22e4·29-s − 1.00e5·30-s − 3.06e5·31-s − 7.29e4·32-s + ⋯ |
| L(s) = 1 | + 1.76·2-s − 0.427·3-s + 0.937·4-s + 0.894·5-s − 0.756·6-s − 0.441·8-s + 0.361·9-s + 1.58·10-s + 1.02·11-s − 0.400·12-s − 0.446·13-s − 0.382·15-s − 0.707·16-s + 1.34·17-s + 0.638·18-s − 1.29·19-s + 0.838·20-s + 1.81·22-s − 2.12·23-s + 0.189·24-s + 3/5·25-s − 0.789·26-s − 0.658·27-s − 0.550·29-s − 0.676·30-s − 1.84·31-s − 0.393·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 60025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60025 ^{s/2} \, \Gamma_{\C}(s+7/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(2.371757829\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.371757829\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | $C_1$ | \( ( 1 - p^{3} T )^{2} \) |
| 7 | | \( 1 \) |
| good | 2 | $D_{4}$ | \( 1 - 5 p^{2} T + 35 p^{3} T^{2} - 5 p^{9} T^{3} + p^{14} T^{4} \) |
| 3 | $D_{4}$ | \( 1 + 20 T - 130 p T^{2} + 20 p^{7} T^{3} + p^{14} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 4544 T + 31976326 T^{2} - 4544 p^{7} T^{3} + p^{14} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 3540 T + 100535470 T^{2} + 3540 p^{7} T^{3} + p^{14} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 27340 T + 901005190 T^{2} - 27340 p^{7} T^{3} + p^{14} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 2040 p T + 113449762 p T^{2} + 2040 p^{8} T^{3} + p^{14} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 124140 T + 10649684530 T^{2} + 124140 p^{7} T^{3} + p^{14} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 72260 T + 6846819118 T^{2} + 72260 p^{7} T^{3} + p^{14} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 306824 T + 77964629966 T^{2} + 306824 p^{7} T^{3} + p^{14} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 123020 T + 144088599870 T^{2} + 123020 p^{7} T^{3} + p^{14} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 264364 T + 161786388886 T^{2} + 264364 p^{7} T^{3} + p^{14} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 423300 T + 446651231050 T^{2} - 423300 p^{7} T^{3} + p^{14} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 105460 T + 858715356610 T^{2} - 105460 p^{7} T^{3} + p^{14} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 2391580 T + 3562552504510 T^{2} + 2391580 p^{7} T^{3} + p^{14} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 1120120 T + 1362334883638 T^{2} - 1120120 p^{7} T^{3} + p^{14} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 2257044 T + 5613447576526 T^{2} + 2257044 p^{7} T^{3} + p^{14} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 4516460 T + 16742087664890 T^{2} - 4516460 p^{7} T^{3} + p^{14} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 621784 T + 17914494152446 T^{2} - 621784 p^{7} T^{3} + p^{14} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 4569060 T + 23424949855030 T^{2} + 4569060 p^{7} T^{3} + p^{14} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 4333040 T + 330830231042 p T^{2} - 4333040 p^{7} T^{3} + p^{14} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 9793020 T + 59971104320890 T^{2} - 9793020 p^{7} T^{3} + p^{14} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 6025620 T + 89865866149558 T^{2} + 6025620 p^{7} T^{3} + p^{14} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 4609540 T + 142930351581510 T^{2} + 4609540 p^{7} T^{3} + p^{14} T^{4} \) |
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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
−11.16514867656644033091514826898, −10.75474325417251231483165750204, −10.07751155365651647707130193715, −9.805559555848707768119938419303, −9.183587139701274221585739958974, −8.886478375569968658716521894938, −7.77009349332815763447308605365, −7.74533795078474453699793217054, −6.53954926636229877712461759055, −6.50322303913740746200718516487, −5.72439854173918350448533171672, −5.51384799984159806340256428432, −5.00937556091010031356489169891, −4.27374646704851205212993661776, −3.85170207970975579264029762638, −3.56471018238397491699417700918, −2.59042872129000343311679403117, −1.76982852257489053666622982073, −1.50602363715194693917241115268, −0.25530187865794903430457651483,
0.25530187865794903430457651483, 1.50602363715194693917241115268, 1.76982852257489053666622982073, 2.59042872129000343311679403117, 3.56471018238397491699417700918, 3.85170207970975579264029762638, 4.27374646704851205212993661776, 5.00937556091010031356489169891, 5.51384799984159806340256428432, 5.72439854173918350448533171672, 6.50322303913740746200718516487, 6.53954926636229877712461759055, 7.74533795078474453699793217054, 7.77009349332815763447308605365, 8.886478375569968658716521894938, 9.183587139701274221585739958974, 9.805559555848707768119938419303, 10.07751155365651647707130193715, 10.75474325417251231483165750204, 11.16514867656644033091514826898
