| L(s) = 1 | + (−31.9 + 0.770i)2-s − 80.5·3-s + (1.02e3 − 49.2i)4-s + (2.57e3 − 62.0i)6-s − 345.·7-s + (−3.26e4 + 2.36e3i)8-s − 5.25e4·9-s + 1.67e5i·11-s + (−8.23e4 + 3.97e3i)12-s + 9.56e4i·13-s + (1.10e4 − 265. i)14-s + (1.04e6 − 1.00e5i)16-s − 2.26e6i·17-s + (1.68e6 − 4.04e4i)18-s + 1.35e6i·19-s + ⋯ |
| L(s) = 1 | + (−0.999 + 0.0240i)2-s − 0.331·3-s + (0.998 − 0.0481i)4-s + (0.331 − 0.00798i)6-s − 0.0205·7-s + (−0.997 + 0.0721i)8-s − 0.890·9-s + 1.03i·11-s + (−0.331 + 0.0159i)12-s + 0.257i·13-s + (0.0205 − 0.000494i)14-s + (0.995 − 0.0961i)16-s − 1.59i·17-s + (0.889 − 0.0214i)18-s + 0.547i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.489 + 0.871i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (0.489 + 0.871i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{11}{2})\) |
\(\approx\) |
\(0.465791 - 0.272602i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.465791 - 0.272602i\) |
| \(L(6)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (31.9 - 0.770i)T \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + 80.5T + 5.90e4T^{2} \) |
| 7 | \( 1 + 345.T + 2.82e8T^{2} \) |
| 11 | \( 1 - 1.67e5iT - 2.59e10T^{2} \) |
| 13 | \( 1 - 9.56e4iT - 1.37e11T^{2} \) |
| 17 | \( 1 + 2.26e6iT - 2.01e12T^{2} \) |
| 19 | \( 1 - 1.35e6iT - 6.13e12T^{2} \) |
| 23 | \( 1 + 7.72e6T + 4.14e13T^{2} \) |
| 29 | \( 1 - 6.87e6T + 4.20e14T^{2} \) |
| 31 | \( 1 - 4.09e7iT - 8.19e14T^{2} \) |
| 37 | \( 1 - 4.60e7iT - 4.80e15T^{2} \) |
| 41 | \( 1 - 1.22e7T + 1.34e16T^{2} \) |
| 43 | \( 1 + 2.14e8T + 2.16e16T^{2} \) |
| 47 | \( 1 - 1.55e8T + 5.25e16T^{2} \) |
| 53 | \( 1 + 5.32e7iT - 1.74e17T^{2} \) |
| 59 | \( 1 - 9.05e8iT - 5.11e17T^{2} \) |
| 61 | \( 1 - 7.30e8T + 7.13e17T^{2} \) |
| 67 | \( 1 + 5.67e8T + 1.82e18T^{2} \) |
| 71 | \( 1 + 2.07e9iT - 3.25e18T^{2} \) |
| 73 | \( 1 + 3.04e9iT - 4.29e18T^{2} \) |
| 79 | \( 1 + 2.49e9iT - 9.46e18T^{2} \) |
| 83 | \( 1 - 6.00e9T + 1.55e19T^{2} \) |
| 89 | \( 1 + 5.14e9T + 3.11e19T^{2} \) |
| 97 | \( 1 + 6.01e9iT - 7.37e19T^{2} \) |
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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
−11.71397874637279997374318478954, −10.45082121453243302125206329644, −9.572583108113399725328767532264, −8.486426571872996137341249100354, −7.34435485211708386101868685864, −6.31144762306225359185124044439, −4.98454075918632260919236550651, −3.03733497868753395343774592998, −1.76892459136747130788134354174, −0.28480718209621580704298685152,
0.71316262267130794091690735643, 2.23096873811804862884286382959, 3.57870209000960478205048708134, 5.68321463302747683257522646015, 6.40388784901125239046326041426, 8.020466641559470373469747817058, 8.625757664573642528959944218697, 9.951902206714067911284193335585, 10.97466246025743729618169291678, 11.63534202349298878003407422534
