| L(s) = 1 | + 7.17·3-s + (−1.28 − 55.8i)5-s − 146. i·7-s − 191.·9-s + 42.4i·11-s − 605.·13-s + (−9.21 − 401. i)15-s + 409. i·17-s + 2.09e3i·19-s − 1.05e3i·21-s + 3.04e3i·23-s + (−3.12e3 + 143. i)25-s − 3.11e3·27-s − 4.59e3i·29-s + 5.11e3·31-s + ⋯ |
| L(s) = 1 | + 0.460·3-s + (−0.0229 − 0.999i)5-s − 1.13i·7-s − 0.787·9-s + 0.105i·11-s − 0.993·13-s + (−0.0105 − 0.460i)15-s + 0.343i·17-s + 1.33i·19-s − 0.521i·21-s + 1.20i·23-s + (−0.998 + 0.0459i)25-s − 0.823·27-s − 1.01i·29-s + 0.956·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.973 - 0.228i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.973 - 0.228i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.4965349858\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4965349858\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (1.28 + 55.8i)T \) |
| good | 3 | \( 1 - 7.17T + 243T^{2} \) |
| 7 | \( 1 + 146. iT - 1.68e4T^{2} \) |
| 11 | \( 1 - 42.4iT - 1.61e5T^{2} \) |
| 13 | \( 1 + 605.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 409. iT - 1.41e6T^{2} \) |
| 19 | \( 1 - 2.09e3iT - 2.47e6T^{2} \) |
| 23 | \( 1 - 3.04e3iT - 6.43e6T^{2} \) |
| 29 | \( 1 + 4.59e3iT - 2.05e7T^{2} \) |
| 31 | \( 1 - 5.11e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.12e4T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.10e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 8.41e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 3.97e3iT - 2.29e8T^{2} \) |
| 53 | \( 1 + 3.82e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 3.66e4iT - 7.14e8T^{2} \) |
| 61 | \( 1 - 3.18e3iT - 8.44e8T^{2} \) |
| 67 | \( 1 + 4.18e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 7.21e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 4.82e4iT - 2.07e9T^{2} \) |
| 79 | \( 1 - 3.94e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 4.65e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 3.82e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 9.59e4iT - 8.58e9T^{2} \) |
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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
−11.59634363567612788301468445318, −10.23217914159288197868312583822, −9.422728238493584889186569884865, −8.207207132142496300947298631797, −7.54007133031609032594928643228, −5.91325841403226726701167510832, −4.64920123946776310655281713733, −3.48155375005024466552514624134, −1.69423344633219158541950613606, −0.14272689846127961568630516247,
2.44297731489683975836598598785, 2.98611606675135172522960798544, 4.92244761118949077608749578678, 6.18728337706269398121011623725, 7.27416483555029577900888789709, 8.534309487243571793782101038946, 9.321457336565315484638764036344, 10.56033459362954346360013593332, 11.55650367252931668834753084384, 12.36438163753357163080461474441
