| L(s) = 1 | + 4·2-s + 10·3-s + 84·5-s + 40·6-s − 64·8-s + 243·9-s + 336·10-s + 336·11-s − 1.16e3·13-s + 840·15-s − 256·16-s − 1.45e3·17-s + 972·18-s + 470·19-s + 1.34e3·22-s + 4.20e3·23-s − 640·24-s + 3.12e3·25-s − 4.67e3·26-s + 6.29e3·27-s + 9.73e3·29-s + 3.36e3·30-s − 7.37e3·31-s + 3.36e3·33-s − 5.83e3·34-s − 1.43e4·37-s + 1.88e3·38-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.641·3-s + 1.50·5-s + 0.453·6-s − 0.353·8-s + 9-s + 1.06·10-s + 0.837·11-s − 1.91·13-s + 0.963·15-s − 1/4·16-s − 1.22·17-s + 0.707·18-s + 0.298·19-s + 0.592·22-s + 1.65·23-s − 0.226·24-s + 25-s − 1.35·26-s + 1.66·27-s + 2.14·29-s + 0.681·30-s − 1.37·31-s + 0.537·33-s − 0.865·34-s − 1.72·37-s + 0.211·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9604 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9604 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(6.407170016\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.407170016\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_2$ | \( 1 - p^{2} T + p^{4} T^{2} \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 - 10 T - 143 T^{2} - 10 p^{5} T^{3} + p^{10} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 84 T + 3931 T^{2} - 84 p^{5} T^{3} + p^{10} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 336 T - 48155 T^{2} - 336 p^{5} T^{3} + p^{10} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 584 T + p^{5} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 1458 T + 705907 T^{2} + 1458 p^{5} T^{3} + p^{10} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 470 T - 2255199 T^{2} - 470 p^{5} T^{3} + p^{10} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 4200 T + 11203657 T^{2} - 4200 p^{5} T^{3} + p^{10} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 4866 T + p^{5} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 7372 T + 25717233 T^{2} + 7372 p^{5} T^{3} + p^{10} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 14330 T + 136004943 T^{2} + 14330 p^{5} T^{3} + p^{10} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 6222 T + p^{5} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 3704 T + p^{5} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 1812 T - 226061663 T^{2} + 1812 p^{5} T^{3} + p^{10} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 37242 T + 968771071 T^{2} - 37242 p^{5} T^{3} + p^{10} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 34302 T + 461702905 T^{2} - 34302 p^{5} T^{3} + p^{10} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 24476 T - 245521725 T^{2} - 24476 p^{5} T^{3} + p^{10} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 17452 T - 1045552803 T^{2} - 17452 p^{5} T^{3} + p^{10} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 28224 T + p^{5} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 3602 T - 2060097189 T^{2} - 3602 p^{5} T^{3} + p^{10} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 42872 T - 1239048015 T^{2} + 42872 p^{5} T^{3} + p^{10} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 35202 T + p^{5} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 26730 T - 4869566549 T^{2} - 26730 p^{5} T^{3} + p^{10} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 16978 T + p^{5} T^{2} )^{2} \) |
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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
−13.59211709189217075425446633789, −12.60107257665548804391872865371, −12.56550024395362911986304780062, −11.89767738394204554095450382699, −11.09129105595796595572184127104, −10.22749580723280521425237582562, −10.07722999777935460457955803517, −9.447447841024917438386079117027, −8.802608064368352772233621908433, −8.647096181778386197190966858699, −7.30263430629295054783955835797, −6.78169687999713946472967512441, −6.64656469033934855397053644493, −5.35684189062917643696109525949, −5.03261430322521054081143665524, −4.40404859765846826812380961120, −3.39861724708033735751536243205, −2.50529295948567086069025903300, −2.00664590801002533996666942040, −0.884845058604856041388485971121,
0.884845058604856041388485971121, 2.00664590801002533996666942040, 2.50529295948567086069025903300, 3.39861724708033735751536243205, 4.40404859765846826812380961120, 5.03261430322521054081143665524, 5.35684189062917643696109525949, 6.64656469033934855397053644493, 6.78169687999713946472967512441, 7.30263430629295054783955835797, 8.647096181778386197190966858699, 8.802608064368352772233621908433, 9.447447841024917438386079117027, 10.07722999777935460457955803517, 10.22749580723280521425237582562, 11.09129105595796595572184127104, 11.89767738394204554095450382699, 12.56550024395362911986304780062, 12.60107257665548804391872865371, 13.59211709189217075425446633789
