| L(s) = 1 | + (−1 + i)3-s + (3 + 3i)7-s + i·9-s + 2i·11-s + (−3 − 3i)13-s + (−1 + i)17-s + 4·19-s − 6·21-s + (1 − i)23-s + (−4 − 4i)27-s + 10i·31-s + (−2 − 2i)33-s + (1 − i)37-s + 6·39-s − 10·41-s + ⋯ |
| L(s) = 1 | + (−0.577 + 0.577i)3-s + (1.13 + 1.13i)7-s + 0.333i·9-s + 0.603i·11-s + (−0.832 − 0.832i)13-s + (−0.242 + 0.242i)17-s + 0.917·19-s − 1.30·21-s + (0.208 − 0.208i)23-s + (−0.769 − 0.769i)27-s + 1.79i·31-s + (−0.348 − 0.348i)33-s + (0.164 − 0.164i)37-s + 0.960·39-s − 1.56·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.525 - 0.850i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.525 - 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.573689 + 1.02897i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.573689 + 1.02897i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + (1 - i)T - 3iT^{2} \) |
| 7 | \( 1 + (-3 - 3i)T + 7iT^{2} \) |
| 11 | \( 1 - 2iT - 11T^{2} \) |
| 13 | \( 1 + (3 + 3i)T + 13iT^{2} \) |
| 17 | \( 1 + (1 - i)T - 17iT^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 23 | \( 1 + (-1 + i)T - 23iT^{2} \) |
| 29 | \( 1 - 29T^{2} \) |
| 31 | \( 1 - 10iT - 31T^{2} \) |
| 37 | \( 1 + (-1 + i)T - 37iT^{2} \) |
| 41 | \( 1 + 10T + 41T^{2} \) |
| 43 | \( 1 + (5 - 5i)T - 43iT^{2} \) |
| 47 | \( 1 + (-3 - 3i)T + 47iT^{2} \) |
| 53 | \( 1 + (-5 - 5i)T + 53iT^{2} \) |
| 59 | \( 1 + 12T + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 + (-1 - i)T + 67iT^{2} \) |
| 71 | \( 1 - 2iT - 71T^{2} \) |
| 73 | \( 1 + (1 + i)T + 73iT^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 + (5 - 5i)T - 83iT^{2} \) |
| 89 | \( 1 + 16iT - 89T^{2} \) |
| 97 | \( 1 + (-3 + 3i)T - 97iT^{2} \) |
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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
−10.53801898130760231602550792017, −9.881633541073524618857510399796, −8.845966807037717208827539248878, −8.055300510133531601149933734628, −7.20776009761738679295576243310, −5.84321312207294119439341714655, −5.04279350927061140336413948223, −4.70724086947760161543118487461, −2.98583320373619828613211231407, −1.78704947084357478927538519404,
0.64506924347125637965437535748, 1.86629275310895654054021106238, 3.58560007532189503648850603494, 4.63355241801052051179115717180, 5.51986834619945442717582828569, 6.68522468942190639094060403261, 7.29577776853698128231862495224, 8.043369381561794008056577988471, 9.177523187138729099815204687783, 10.04601612046077669039541069461
