| L(s) = 1 | + (−3.26 + 5.65i)2-s + (122. + 212. i)3-s + (234. + 406. i)4-s + (−940. − 1.62e3i)5-s − 1.60e3·6-s + (3.05e3 + 5.29e3i)7-s − 6.40e3·8-s + (−2.03e4 + 3.52e4i)9-s + 1.22e4·10-s − 5.05e4·11-s + (−5.76e4 + 9.98e4i)12-s + (3.45e4 + 5.99e4i)13-s − 3.98e4·14-s + (2.31e5 − 4.00e5i)15-s + (−9.92e4 + 1.71e5i)16-s + (1.89e5 − 3.28e5i)17-s + ⋯ |
| L(s) = 1 | + (−0.144 + 0.249i)2-s + (0.875 + 1.51i)3-s + (0.458 + 0.794i)4-s + (−0.673 − 1.16i)5-s − 0.505·6-s + (0.480 + 0.833i)7-s − 0.552·8-s + (−1.03 + 1.78i)9-s + 0.388·10-s − 1.04·11-s + (−0.802 + 1.39i)12-s + (0.335 + 0.581i)13-s − 0.277·14-s + (1.17 − 2.04i)15-s + (−0.378 + 0.655i)16-s + (0.550 − 0.953i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.990 + 0.138i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.990 + 0.138i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.121840 - 1.75224i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.121840 - 1.75224i\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 37 | \( 1 + (-5.61e6 - 9.91e6i)T \) |
| good | 2 | \( 1 + (3.26 - 5.65i)T + (-256 - 443. i)T^{2} \) |
| 3 | \( 1 + (-122. - 212. i)T + (-9.84e3 + 1.70e4i)T^{2} \) |
| 5 | \( 1 + (940. + 1.62e3i)T + (-9.76e5 + 1.69e6i)T^{2} \) |
| 7 | \( 1 + (-3.05e3 - 5.29e3i)T + (-2.01e7 + 3.49e7i)T^{2} \) |
| 11 | \( 1 + 5.05e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + (-3.45e4 - 5.99e4i)T + (-5.30e9 + 9.18e9i)T^{2} \) |
| 17 | \( 1 + (-1.89e5 + 3.28e5i)T + (-5.92e10 - 1.02e11i)T^{2} \) |
| 19 | \( 1 + (-1.05e5 - 1.82e5i)T + (-1.61e11 + 2.79e11i)T^{2} \) |
| 23 | \( 1 + 1.99e6T + 1.80e12T^{2} \) |
| 29 | \( 1 - 5.01e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 3.79e6T + 2.64e13T^{2} \) |
| 41 | \( 1 + (4.88e5 + 8.46e5i)T + (-1.63e14 + 2.83e14i)T^{2} \) |
| 43 | \( 1 - 2.30e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + 3.79e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + (2.95e7 - 5.11e7i)T + (-1.64e15 - 2.85e15i)T^{2} \) |
| 59 | \( 1 + (-3.62e7 + 6.28e7i)T + (-4.33e15 - 7.50e15i)T^{2} \) |
| 61 | \( 1 + (-1.54e7 - 2.68e7i)T + (-5.84e15 + 1.01e16i)T^{2} \) |
| 67 | \( 1 + (-8.15e7 - 1.41e8i)T + (-1.36e16 + 2.35e16i)T^{2} \) |
| 71 | \( 1 + (-1.89e8 - 3.28e8i)T + (-2.29e16 + 3.97e16i)T^{2} \) |
| 73 | \( 1 - 3.39e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + (-5.20e7 - 9.01e7i)T + (-5.99e16 + 1.03e17i)T^{2} \) |
| 83 | \( 1 + (3.47e8 - 6.01e8i)T + (-9.34e16 - 1.61e17i)T^{2} \) |
| 89 | \( 1 + (5.46e8 - 9.46e8i)T + (-1.75e17 - 3.03e17i)T^{2} \) |
| 97 | \( 1 + 1.50e9T + 7.60e17T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−15.52637835715619337807192713375, −14.09739308261999547595294126028, −12.42377708328408160797832121912, −11.38304514446653167310148148499, −9.671803673295351906529910537945, −8.469661145186351886287369258800, −8.043692764340695170185973084644, −5.20729735733854634685618088737, −4.02888001709711222150677273951, −2.60257442070245453125970742537,
0.57914290813355643107442210912, 2.01063980816780290105673768309, 3.24485766711607984312930436380, 6.18814325489379125188170707812, 7.39431747683707405525325249330, 8.074442150278813420898602520710, 10.28108909057932757830409870036, 11.20896788366684024399650055499, 12.58601476549209206115603604500, 13.96762396418458150576059623471
