| L(s) = 1 | + 9.97i·5-s − 2.20·7-s − 9·9-s + 20.3i·11-s + 18.9·17-s + 19i·19-s + 34.8·23-s − 74.4·25-s − 22.0i·35-s + 53.8i·43-s − 89.7i·45-s − 36.6·47-s − 44.1·49-s − 203.·55-s − 121. i·61-s + ⋯ |
| L(s) = 1 | + 1.99i·5-s − 0.315·7-s − 9-s + 1.85i·11-s + 1.11·17-s + i·19-s + 1.51·23-s − 2.97·25-s − 0.629i·35-s + 1.25i·43-s − 1.99i·45-s − 0.779·47-s − 0.900·49-s − 3.69·55-s − 1.99i·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.965 + 0.258i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.965 + 0.258i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.206527529\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.206527529\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 - 19iT \) |
| good | 3 | \( 1 + 9T^{2} \) |
| 5 | \( 1 - 9.97iT - 25T^{2} \) |
| 7 | \( 1 + 2.20T + 49T^{2} \) |
| 11 | \( 1 - 20.3iT - 121T^{2} \) |
| 13 | \( 1 + 169T^{2} \) |
| 17 | \( 1 - 18.9T + 289T^{2} \) |
| 23 | \( 1 - 34.8T + 529T^{2} \) |
| 29 | \( 1 + 841T^{2} \) |
| 31 | \( 1 - 961T^{2} \) |
| 37 | \( 1 + 1.36e3T^{2} \) |
| 41 | \( 1 - 1.68e3T^{2} \) |
| 43 | \( 1 - 53.8iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 36.6T + 2.20e3T^{2} \) |
| 53 | \( 1 + 2.80e3T^{2} \) |
| 59 | \( 1 + 3.48e3T^{2} \) |
| 61 | \( 1 + 121. iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 4.48e3T^{2} \) |
| 71 | \( 1 - 5.04e3T^{2} \) |
| 73 | \( 1 - 112.T + 5.32e3T^{2} \) |
| 79 | \( 1 - 6.24e3T^{2} \) |
| 83 | \( 1 + 90iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 7.92e3T^{2} \) |
| 97 | \( 1 - 9.40e3T^{2} \) |
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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
−9.872787970681359247385977638385, −9.602339350232380378747791736142, −8.074405922059306094630345550772, −7.43406150447864433660435471638, −6.70615502578127658134614782197, −6.03560398861788176653244656616, −4.93150446058590892820945172269, −3.52598614014140722800648326154, −2.95886932010051714172849050168, −1.92029022163731088158712209538,
0.39943775068056462498006173879, 1.10254222728335197246984775478, 2.86609283514675017240011422744, 3.78987573426375244784324605424, 5.17332904108049297647425610927, 5.39819179558062888187421558177, 6.36848734114160686910589548398, 7.77254119258894073448666182103, 8.575381718556716625853320693702, 8.865899633240250296860038545929
