| L(s) = 1 | + (−4.21 + 4.21i)7-s + 152.·11-s + (136. + 136. i)13-s + (348. − 348. i)17-s + 527. i·19-s + (−70.5 − 70.5i)23-s + 68.9i·29-s − 1.37e3·31-s + (1.00e3 − 1.00e3i)37-s − 663.·41-s + (−1.63e3 − 1.63e3i)43-s + (438. − 438. i)47-s + 2.36e3i·49-s + (712. + 712. i)53-s + 2.91e3i·59-s + ⋯ |
| L(s) = 1 | + (−0.0859 + 0.0859i)7-s + 1.26·11-s + (0.808 + 0.808i)13-s + (1.20 − 1.20i)17-s + 1.46i·19-s + (−0.133 − 0.133i)23-s + 0.0820i·29-s − 1.42·31-s + (0.732 − 0.732i)37-s − 0.394·41-s + (−0.882 − 0.882i)43-s + (0.198 − 0.198i)47-s + 0.985i·49-s + (0.253 + 0.253i)53-s + 0.838i·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.850 - 0.525i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.850 - 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(2.521529731\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.521529731\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + (4.21 - 4.21i)T - 2.40e3iT^{2} \) |
| 11 | \( 1 - 152.T + 1.46e4T^{2} \) |
| 13 | \( 1 + (-136. - 136. i)T + 2.85e4iT^{2} \) |
| 17 | \( 1 + (-348. + 348. i)T - 8.35e4iT^{2} \) |
| 19 | \( 1 - 527. iT - 1.30e5T^{2} \) |
| 23 | \( 1 + (70.5 + 70.5i)T + 2.79e5iT^{2} \) |
| 29 | \( 1 - 68.9iT - 7.07e5T^{2} \) |
| 31 | \( 1 + 1.37e3T + 9.23e5T^{2} \) |
| 37 | \( 1 + (-1.00e3 + 1.00e3i)T - 1.87e6iT^{2} \) |
| 41 | \( 1 + 663.T + 2.82e6T^{2} \) |
| 43 | \( 1 + (1.63e3 + 1.63e3i)T + 3.41e6iT^{2} \) |
| 47 | \( 1 + (-438. + 438. i)T - 4.87e6iT^{2} \) |
| 53 | \( 1 + (-712. - 712. i)T + 7.89e6iT^{2} \) |
| 59 | \( 1 - 2.91e3iT - 1.21e7T^{2} \) |
| 61 | \( 1 - 1.39e3T + 1.38e7T^{2} \) |
| 67 | \( 1 + (-1.69e3 + 1.69e3i)T - 2.01e7iT^{2} \) |
| 71 | \( 1 - 3.28e3T + 2.54e7T^{2} \) |
| 73 | \( 1 + (-1.46e3 - 1.46e3i)T + 2.83e7iT^{2} \) |
| 79 | \( 1 + 3.38e3iT - 3.89e7T^{2} \) |
| 83 | \( 1 + (-8.69e3 - 8.69e3i)T + 4.74e7iT^{2} \) |
| 89 | \( 1 + 4.27e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 + (7.21e3 - 7.21e3i)T - 8.85e7iT^{2} \) |
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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
−9.468202200074544415512256984386, −8.965109115775705958076474199822, −7.904850669003738101792265578070, −7.04046519079992018202696178645, −6.16394017922120662177095538667, −5.35545576586241721526005651728, −4.04095469321828093291064677929, −3.43886731514870211870548058333, −1.91273604281966358438747810593, −0.947024518250591364563911860194,
0.71082360777867226241885790656, 1.69486513522751159386911479279, 3.23536287726445817371722407754, 3.89334367149474427803149832548, 5.13001118673063739193815361221, 6.08310278566720447573786765773, 6.79906202296384049824718869258, 7.87474977641850091940826706391, 8.621962064564712063183030150273, 9.462985032301590018207861971671
