| L(s) = 1 | + (2.82 + i)3-s + i·7-s + (7.00 + 5.65i)9-s + 8.48i·11-s + 15i·13-s − 19.7·17-s − 23·19-s + (−1 + 2.82i)21-s − 2.82·23-s + (14.1 + 23.0i)27-s + 25.4i·29-s − 33·31-s + (−8.48 + 24i)33-s − 66i·37-s + (−15 + 42.4i)39-s + ⋯ |
| L(s) = 1 | + (0.942 + 0.333i)3-s + 0.142i·7-s + (0.777 + 0.628i)9-s + 0.771i·11-s + 1.15i·13-s − 1.16·17-s − 1.21·19-s + (−0.0476 + 0.134i)21-s − 0.122·23-s + (0.523 + 0.851i)27-s + 0.877i·29-s − 1.06·31-s + (−0.257 + 0.727i)33-s − 1.78i·37-s + (−0.384 + 1.08i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.694 - 0.719i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.694 - 0.719i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.802763652\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.802763652\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-2.82 - i)T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - iT - 49T^{2} \) |
| 11 | \( 1 - 8.48iT - 121T^{2} \) |
| 13 | \( 1 - 15iT - 169T^{2} \) |
| 17 | \( 1 + 19.7T + 289T^{2} \) |
| 19 | \( 1 + 23T + 361T^{2} \) |
| 23 | \( 1 + 2.82T + 529T^{2} \) |
| 29 | \( 1 - 25.4iT - 841T^{2} \) |
| 31 | \( 1 + 33T + 961T^{2} \) |
| 37 | \( 1 + 66iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 36.7iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 7iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 45.2T + 2.20e3T^{2} \) |
| 53 | \( 1 - 36.7T + 2.80e3T^{2} \) |
| 59 | \( 1 - 101. iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 39T + 3.72e3T^{2} \) |
| 67 | \( 1 - 113iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 25.4iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 58iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 70T + 6.24e3T^{2} \) |
| 83 | \( 1 + 152.T + 6.88e3T^{2} \) |
| 89 | \( 1 - 90.5iT - 7.92e3T^{2} \) |
| 97 | \( 1 - iT - 9.40e3T^{2} \) |
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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
−9.690043381559262774008280623274, −8.888096937119546819570430762523, −8.602058268967815906872857036226, −7.22465025988918728102125218052, −6.91942951764868766190208978250, −5.54608549277465975191815446576, −4.34100468072992717919547037759, −3.97892338604426368140308358315, −2.45320062288465307626537305034, −1.84059687844131576518589366642,
0.42846827802913743869167884567, 1.89640581715012485815352247052, 2.92127175643012196596575310802, 3.80176572498985898740282173210, 4.82643302652638781520783789945, 6.12160939601965118047795796406, 6.75482483267630954374076762635, 7.892090676406081093289612985461, 8.331099159413797665393085951907, 9.088201567212220164445678804476
