L(s) = 1 | + (1.40 + 0.199i)2-s + (0.520 + 1.65i)3-s + (1.92 + 0.557i)4-s + (0.400 + 2.41i)6-s + 1.92i·7-s + (2.57 + 1.16i)8-s + (−2.45 + 1.72i)9-s − 4.02i·11-s + (0.0792 + 3.46i)12-s + 4.81i·13-s + (−0.383 + 2.69i)14-s + (3.37 + 2.14i)16-s − 5.23i·17-s + (−3.78 + 1.91i)18-s − 0.684·19-s + ⋯ |
L(s) = 1 | + (0.990 + 0.140i)2-s + (0.300 + 0.953i)3-s + (0.960 + 0.278i)4-s + (0.163 + 0.986i)6-s + 0.728i·7-s + (0.911 + 0.411i)8-s + (−0.819 + 0.573i)9-s − 1.21i·11-s + (0.0228 + 0.999i)12-s + 1.33i·13-s + (−0.102 + 0.721i)14-s + (0.844 + 0.535i)16-s − 1.26i·17-s + (−0.891 + 0.452i)18-s − 0.157·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.118 - 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.118 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.18783 + 1.94312i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.18783 + 1.94312i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.40 - 0.199i)T \) |
| 3 | \( 1 + (-0.520 - 1.65i)T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 - 1.92iT - 7T^{2} \) |
| 11 | \( 1 + 4.02iT - 11T^{2} \) |
| 13 | \( 1 - 4.81iT - 13T^{2} \) |
| 17 | \( 1 + 5.23iT - 17T^{2} \) |
| 19 | \( 1 + 0.684T + 19T^{2} \) |
| 23 | \( 1 + 1.72T + 23T^{2} \) |
| 29 | \( 1 - 6.99T + 29T^{2} \) |
| 31 | \( 1 - 4.23iT - 31T^{2} \) |
| 37 | \( 1 + 9.83iT - 37T^{2} \) |
| 41 | \( 1 + 3.44iT - 41T^{2} \) |
| 43 | \( 1 + 1.04T + 43T^{2} \) |
| 47 | \( 1 + 7.55T + 47T^{2} \) |
| 53 | \( 1 - 4.08T + 53T^{2} \) |
| 59 | \( 1 - 0.994iT - 59T^{2} \) |
| 61 | \( 1 + 3.16iT - 61T^{2} \) |
| 67 | \( 1 - 14.8T + 67T^{2} \) |
| 71 | \( 1 + 9.28T + 71T^{2} \) |
| 73 | \( 1 + 11.2T + 73T^{2} \) |
| 79 | \( 1 - 9.25iT - 79T^{2} \) |
| 83 | \( 1 + 7.15iT - 83T^{2} \) |
| 89 | \( 1 + 0.829iT - 89T^{2} \) |
| 97 | \( 1 - 1.45T + 97T^{2} \) |
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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
−11.15653767000832188252493096206, −10.09020502832102981682992229263, −9.002411876680066694946910470465, −8.418949618699232275219283297419, −7.10768028383281617524175151019, −6.04787104788268031515726513921, −5.22902000849961253009020622285, −4.33003540876792570189455328890, −3.28719310343016083032386289689, −2.33886626628842526059723728210,
1.32225889162674717273138569481, 2.57365498917956308419530990063, 3.70336397103580035345157576727, 4.81040714835259061982445728844, 6.02541338116399677480620773075, 6.75302272048289866504562830634, 7.67150960426230306251013047733, 8.284368504781216376885508430684, 9.984042633168194835405343611089, 10.47011681239834513386292070736