| L(s) = 1 | + (1 − i)2-s − 2i·4-s + (3 − 3i)7-s + (−2 − 2i)8-s − 6i·14-s − 4·16-s + (1 + i)23-s + (−6 − 6i)28-s − 6i·29-s + (−4 + 4i)32-s − 12·41-s + (9 + 9i)43-s + 2·46-s + (7 − 7i)47-s − 11i·49-s + ⋯ |
| L(s) = 1 | + (0.707 − 0.707i)2-s − i·4-s + (1.13 − 1.13i)7-s + (−0.707 − 0.707i)8-s − 1.60i·14-s − 16-s + (0.208 + 0.208i)23-s + (−1.13 − 1.13i)28-s − 1.11i·29-s + (−0.707 + 0.707i)32-s − 1.87·41-s + (1.37 + 1.37i)43-s + 0.294·46-s + (1.02 − 1.02i)47-s − 1.57i·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.525 + 0.850i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.525 + 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.16867 - 2.09613i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.16867 - 2.09613i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1 + i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + (-3 + 3i)T - 7iT^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 13 | \( 1 - 13iT^{2} \) |
| 17 | \( 1 + 17iT^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 + (-1 - i)T + 23iT^{2} \) |
| 29 | \( 1 + 6iT - 29T^{2} \) |
| 31 | \( 1 - 31T^{2} \) |
| 37 | \( 1 + 37iT^{2} \) |
| 41 | \( 1 + 12T + 41T^{2} \) |
| 43 | \( 1 + (-9 - 9i)T + 43iT^{2} \) |
| 47 | \( 1 + (-7 + 7i)T - 47iT^{2} \) |
| 53 | \( 1 - 53iT^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 8T + 61T^{2} \) |
| 67 | \( 1 + (-3 + 3i)T - 67iT^{2} \) |
| 71 | \( 1 - 71T^{2} \) |
| 73 | \( 1 - 73iT^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + (-11 - 11i)T + 83iT^{2} \) |
| 89 | \( 1 + 6iT - 89T^{2} \) |
| 97 | \( 1 + 97iT^{2} \) |
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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
−10.16343299952444866717723104132, −9.195762203984399164701767355218, −8.078992412830036582370007384411, −7.24056898474944006872413298440, −6.21313502162534461243512833285, −5.10961798185405734893363398190, −4.40357451549800893699039208209, −3.54007083781514542106327202164, −2.14267014497467420032854894666, −0.966108576971304450391810362719,
1.98528519039804122427485905946, 3.12451912343664057647706000647, 4.41505097194284748975876512516, 5.21858616755767649957089447804, 5.86220419260858020584297175769, 6.95046358721728681067822128429, 7.80471716672948027902564736696, 8.642548864725372871977947213908, 9.109383217348625344383205361102, 10.59099966305573573881295562982
