| L(s) = 1 | + 2-s − 3·3-s − 47·4-s + 50·5-s − 3·6-s − 63·8-s − 333·9-s + 50·10-s − 601·11-s + 141·12-s + 577·13-s − 150·15-s + 1.20e3·16-s − 41·17-s − 333·18-s − 630·19-s − 2.35e3·20-s − 601·22-s − 442·23-s + 189·24-s + 1.87e3·25-s + 577·26-s + 1.29e3·27-s + 5.88e3·29-s − 150·30-s + 396·31-s + 1.69e3·32-s + ⋯ |
| L(s) = 1 | + 0.176·2-s − 0.192·3-s − 1.46·4-s + 0.894·5-s − 0.0340·6-s − 0.348·8-s − 1.37·9-s + 0.158·10-s − 1.49·11-s + 0.282·12-s + 0.946·13-s − 0.172·15-s + 1.17·16-s − 0.0344·17-s − 0.242·18-s − 0.400·19-s − 1.31·20-s − 0.264·22-s − 0.174·23-s + 0.0669·24-s + 3/5·25-s + 0.167·26-s + 0.342·27-s + 1.29·29-s − 0.0304·30-s + 0.0740·31-s + 0.292·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 60025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60025 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 5 | $C_1$ | \( ( 1 - p^{2} T )^{2} \) |
| 7 | | \( 1 \) |
| good | 2 | $D_{4}$ | \( 1 - T + 3 p^{4} T^{2} - p^{5} T^{3} + p^{10} T^{4} \) |
| 3 | $D_{4}$ | \( 1 + p T + 38 p^{2} T^{2} + p^{6} T^{3} + p^{10} T^{4} \) |
| 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 | $D_{4}$ | \( 1 - 96339 T + 16335863448 T^{2} - 96339 p^{5} T^{3} + p^{10} T^{4} \) |
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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
−10.83783081694962912770927663159, −10.46980060444792679932791795511, −10.09595769944624293901500445824, −9.505899271117031055515624773162, −8.922154543215514659881325676034, −8.679938574054879367287235782787, −8.081125606685575868260148092923, −7.930136787509738824111545849704, −6.66435862089440912418136022170, −6.38929769968070287678029298788, −5.53049335053510602963531060603, −5.44120188345495381305532328127, −4.84688985956393295273805530025, −4.29310175416946882270785403421, −3.27707991760327045233119627765, −2.98667903040177048459845321610, −2.06821309323459536533519211105, −1.13917759008544934626375582867, 0, 0,
1.13917759008544934626375582867, 2.06821309323459536533519211105, 2.98667903040177048459845321610, 3.27707991760327045233119627765, 4.29310175416946882270785403421, 4.84688985956393295273805530025, 5.44120188345495381305532328127, 5.53049335053510602963531060603, 6.38929769968070287678029298788, 6.66435862089440912418136022170, 7.930136787509738824111545849704, 8.081125606685575868260148092923, 8.679938574054879367287235782787, 8.922154543215514659881325676034, 9.505899271117031055515624773162, 10.09595769944624293901500445824, 10.46980060444792679932791795511, 10.83783081694962912770927663159
