| L(s) = 1 | + (13.1 − 54.3i)5-s − 146. i·7-s − 191.·11-s + 83.9i·13-s − 2.00e3i·17-s − 677.·19-s − 1.29e3i·23-s + (−2.77e3 − 1.42e3i)25-s − 3.26e3·29-s + 6.15e3·31-s + (−7.97e3 − 1.93e3i)35-s + 1.13e4i·37-s + 1.05e4·41-s + 1.29e4i·43-s + 9.52e3i·47-s + ⋯ |
| L(s) = 1 | + (0.235 − 0.971i)5-s − 1.13i·7-s − 0.476·11-s + 0.137i·13-s − 1.67i·17-s − 0.430·19-s − 0.510i·23-s + (−0.889 − 0.457i)25-s − 0.721·29-s + 1.15·31-s + (−1.10 − 0.266i)35-s + 1.36i·37-s + 0.984·41-s + 1.06i·43-s + 0.628i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.971 - 0.235i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.971 - 0.235i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.9789542521\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9789542521\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-13.1 + 54.3i)T \) |
| good | 7 | \( 1 + 146. iT - 1.68e4T^{2} \) |
| 11 | \( 1 + 191.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 83.9iT - 3.71e5T^{2} \) |
| 17 | \( 1 + 2.00e3iT - 1.41e6T^{2} \) |
| 19 | \( 1 + 677.T + 2.47e6T^{2} \) |
| 23 | \( 1 + 1.29e3iT - 6.43e6T^{2} \) |
| 29 | \( 1 + 3.26e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 6.15e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.13e4iT - 6.93e7T^{2} \) |
| 41 | \( 1 - 1.05e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.29e4iT - 1.47e8T^{2} \) |
| 47 | \( 1 - 9.52e3iT - 2.29e8T^{2} \) |
| 53 | \( 1 + 1.47e4iT - 4.18e8T^{2} \) |
| 59 | \( 1 + 3.82e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 3.58e3T + 8.44e8T^{2} \) |
| 67 | \( 1 + 2.17e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 + 5.13e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 1.33e4iT - 2.07e9T^{2} \) |
| 79 | \( 1 + 1.59e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 5.33e4iT - 3.93e9T^{2} \) |
| 89 | \( 1 + 5.13e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 8.08e4iT - 8.58e9T^{2} \) |
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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
−10.01207530940005361445753088238, −9.310508834837480158474085513808, −8.195415928404818901433990620702, −7.37829713310632837015891330179, −6.26031010831608240616363197264, −4.94791218068937519668620838392, −4.32410838839663302539568818699, −2.79094613460172583837007120632, −1.24029632202559987382340352784, −0.24946305482556709318897617185,
1.86018936068893876835374007957, 2.77465779133640243076292363839, 3.99694823075174844409305265286, 5.61424379653243611839912590533, 6.11728025502932742426093220421, 7.35070485730426737935024600361, 8.336016685209793251408059201910, 9.270656307957154855661539791519, 10.35433232284521948993660210860, 10.94246847714006339053715860432
