L(s) = 1 | − 2-s + (−0.880 − 1.49i)3-s + 4-s + (2.09 + 0.775i)5-s + (0.880 + 1.49i)6-s − 0.657i·7-s − 8-s + (−1.44 + 2.62i)9-s + (−2.09 − 0.775i)10-s − 3.48·11-s + (−0.880 − 1.49i)12-s + 4.74·13-s + 0.657i·14-s + (−0.690 − 3.81i)15-s + 16-s − 3.35i·17-s + ⋯ |
L(s) = 1 | − 0.707·2-s + (−0.508 − 0.861i)3-s + 0.5·4-s + (0.937 + 0.346i)5-s + (0.359 + 0.608i)6-s − 0.248i·7-s − 0.353·8-s + (−0.482 + 0.875i)9-s + (−0.663 − 0.245i)10-s − 1.05·11-s + (−0.254 − 0.430i)12-s + 1.31·13-s + 0.175i·14-s + (−0.178 − 0.983i)15-s + 0.250·16-s − 0.814i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.260 + 0.965i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.260 + 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.834937 - 0.639418i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.834937 - 0.639418i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 + (0.880 + 1.49i)T \) |
| 5 | \( 1 + (-2.09 - 0.775i)T \) |
| 31 | \( 1 + (-5.54 + 0.469i)T \) |
good | 7 | \( 1 + 0.657iT - 7T^{2} \) |
| 11 | \( 1 + 3.48T + 11T^{2} \) |
| 13 | \( 1 - 4.74T + 13T^{2} \) |
| 17 | \( 1 + 3.35iT - 17T^{2} \) |
| 19 | \( 1 - 0.313T + 19T^{2} \) |
| 23 | \( 1 + 6.17iT - 23T^{2} \) |
| 29 | \( 1 - 1.48T + 29T^{2} \) |
| 37 | \( 1 + 4.95T + 37T^{2} \) |
| 41 | \( 1 + 0.355iT - 41T^{2} \) |
| 43 | \( 1 - 4.78T + 43T^{2} \) |
| 47 | \( 1 + 0.472T + 47T^{2} \) |
| 53 | \( 1 + 3.48iT - 53T^{2} \) |
| 59 | \( 1 + 6.24iT - 59T^{2} \) |
| 61 | \( 1 - 3.55iT - 61T^{2} \) |
| 67 | \( 1 + 15.3iT - 67T^{2} \) |
| 71 | \( 1 - 7.71iT - 71T^{2} \) |
| 73 | \( 1 - 3.30T + 73T^{2} \) |
| 79 | \( 1 + 13.9iT - 79T^{2} \) |
| 83 | \( 1 + 7.45iT - 83T^{2} \) |
| 89 | \( 1 - 0.308T + 89T^{2} \) |
| 97 | \( 1 + 10.7iT - 97T^{2} \) |
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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
−10.13144840582183603160405779298, −8.938104453140320677002513363282, −8.215786284542503252846530638055, −7.30665099211687844994798194995, −6.51704338337533458314438204882, −5.88360980067554297598409642623, −4.89638902547447108231228711082, −2.99248248418878811657931586579, −2.06554166574978190149047449280, −0.76051857329157685692609694489,
1.23723672143041703201029938529, 2.71749311627507902488770577463, 3.95922425382982066198355188528, 5.30544771727908399903717342079, 5.80051473435894147491316375380, 6.64295779777053866918888816536, 8.045926738290451751002422863507, 8.771865849101092597973857232106, 9.440775600154574024860487922515, 10.33568074344104878338037089175