| L(s) = 1 | + 20.1·2-s − 81·3-s − 107.·4-s + 625·5-s − 1.62e3·6-s − 2.84e3·7-s − 1.24e4·8-s + 6.56e3·9-s + 1.25e4·10-s − 9.64e4·11-s + 8.72e3·12-s − 1.18e5·13-s − 5.71e4·14-s − 5.06e4·15-s − 1.95e5·16-s − 3.67e5·17-s + 1.31e5·18-s − 1.30e5·19-s − 6.73e4·20-s + 2.30e5·21-s − 1.94e6·22-s + 2.13e5·23-s + 1.00e6·24-s + 3.90e5·25-s − 2.37e6·26-s − 5.31e5·27-s + 3.05e5·28-s + ⋯ |
| L(s) = 1 | + 0.888·2-s − 0.577·3-s − 0.210·4-s + 0.447·5-s − 0.513·6-s − 0.447·7-s − 1.07·8-s + 0.333·9-s + 0.397·10-s − 1.98·11-s + 0.121·12-s − 1.14·13-s − 0.397·14-s − 0.258·15-s − 0.745·16-s − 1.06·17-s + 0.296·18-s − 0.229·19-s − 0.0940·20-s + 0.258·21-s − 1.76·22-s + 0.158·23-s + 0.620·24-s + 0.200·25-s − 1.02·26-s − 0.192·27-s + 0.0940·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 285 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 285 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.4011927078\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4011927078\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 81T \) |
| 5 | \( 1 - 625T \) |
| 19 | \( 1 + 1.30e5T \) |
| good | 2 | \( 1 - 20.1T + 512T^{2} \) |
| 7 | \( 1 + 2.84e3T + 4.03e7T^{2} \) |
| 11 | \( 1 + 9.64e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + 1.18e5T + 1.06e10T^{2} \) |
| 17 | \( 1 + 3.67e5T + 1.18e11T^{2} \) |
| 23 | \( 1 - 2.13e5T + 1.80e12T^{2} \) |
| 29 | \( 1 + 4.06e5T + 1.45e13T^{2} \) |
| 31 | \( 1 - 2.17e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 9.81e6T + 1.29e14T^{2} \) |
| 41 | \( 1 + 1.41e7T + 3.27e14T^{2} \) |
| 43 | \( 1 - 3.97e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + 5.46e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 8.10e7T + 3.29e15T^{2} \) |
| 59 | \( 1 - 2.82e7T + 8.66e15T^{2} \) |
| 61 | \( 1 - 1.30e8T + 1.16e16T^{2} \) |
| 67 | \( 1 + 8.18e7T + 2.72e16T^{2} \) |
| 71 | \( 1 + 2.87e8T + 4.58e16T^{2} \) |
| 73 | \( 1 - 2.53e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 1.21e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 3.80e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 7.94e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 9.23e8T + 7.60e17T^{2} \) |
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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
−10.26630825730414974054352591825, −9.538913030652291672985430100560, −8.315940218183122124595887993617, −7.06435525646744944848339279726, −6.03721121859098707001086294915, −5.12914210484004856600216039570, −4.61326483959738372900405736310, −3.09846983515463609816922146139, −2.22558988025326436626518903809, −0.23500203592126998992320428987,
0.23500203592126998992320428987, 2.22558988025326436626518903809, 3.09846983515463609816922146139, 4.61326483959738372900405736310, 5.12914210484004856600216039570, 6.03721121859098707001086294915, 7.06435525646744944848339279726, 8.315940218183122124595887993617, 9.538913030652291672985430100560, 10.26630825730414974054352591825
