L(s) = 1 | + (0.707 + 0.707i)2-s + (2.10 − 2.10i)3-s + 1.00i·4-s + 2.97·6-s + (0.978 − 0.978i)7-s + (−0.707 + 0.707i)8-s − 5.82i·9-s + 2.43·11-s + (2.10 + 2.10i)12-s + (1.31 − 1.31i)13-s + 1.38·14-s − 1.00·16-s + (−5.11 + 5.11i)17-s + (4.12 − 4.12i)18-s + (3.48 − 2.61i)19-s + ⋯ |
L(s) = 1 | + (0.499 + 0.499i)2-s + (1.21 − 1.21i)3-s + 0.500i·4-s + 1.21·6-s + (0.369 − 0.369i)7-s + (−0.250 + 0.250i)8-s − 1.94i·9-s + 0.733·11-s + (0.606 + 0.606i)12-s + (0.364 − 0.364i)13-s + 0.369·14-s − 0.250·16-s + (−1.23 + 1.23i)17-s + (0.971 − 0.971i)18-s + (0.800 − 0.599i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.827 + 0.561i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.827 + 0.561i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.08414 - 0.946719i\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.08414 - 0.946719i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.707 - 0.707i)T \) |
| 5 | \( 1 \) |
| 19 | \( 1 + (-3.48 + 2.61i)T \) |
good | 3 | \( 1 + (-2.10 + 2.10i)T - 3iT^{2} \) |
| 7 | \( 1 + (-0.978 + 0.978i)T - 7iT^{2} \) |
| 11 | \( 1 - 2.43T + 11T^{2} \) |
| 13 | \( 1 + (-1.31 + 1.31i)T - 13iT^{2} \) |
| 17 | \( 1 + (5.11 - 5.11i)T - 17iT^{2} \) |
| 23 | \( 1 + (1.29 + 1.29i)T + 23iT^{2} \) |
| 29 | \( 1 + 0.448T + 29T^{2} \) |
| 31 | \( 1 + 3.46iT - 31T^{2} \) |
| 37 | \( 1 + (-4.60 - 4.60i)T + 37iT^{2} \) |
| 41 | \( 1 - 6.28iT - 41T^{2} \) |
| 43 | \( 1 + (7.27 + 7.27i)T + 43iT^{2} \) |
| 47 | \( 1 + (-6.15 + 6.15i)T - 47iT^{2} \) |
| 53 | \( 1 + (9.37 - 9.37i)T - 53iT^{2} \) |
| 59 | \( 1 - 5.89T + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 + (4.17 + 4.17i)T + 67iT^{2} \) |
| 71 | \( 1 - 14.6iT - 71T^{2} \) |
| 73 | \( 1 + (-8.04 - 8.04i)T + 73iT^{2} \) |
| 79 | \( 1 - 6.23T + 79T^{2} \) |
| 83 | \( 1 + (11.7 + 11.7i)T + 83iT^{2} \) |
| 89 | \( 1 + 3.31T + 89T^{2} \) |
| 97 | \( 1 + (6.78 + 6.78i)T + 97iT^{2} \) |
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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
−9.640748875435200554252876274478, −8.677096471675763905200366498672, −8.252617002455128538737523310489, −7.39127125134938635412288216886, −6.70394811133014394928842560400, −5.97578183072621767180382463757, −4.45891643439712375220388789713, −3.58259849413627232987541159333, −2.48435346560211660035650272121, −1.32189480380449462382432294794,
1.85873063633012129631043436374, 2.90817302653748105091312481109, 3.78708090404722436445427521065, 4.54860314752194266745500777476, 5.35360459962044689594459739424, 6.66256058423999065592833526485, 7.86200099183794715137397558615, 8.821606069656653182592264052141, 9.342000711395036326212247963946, 9.916071769863927478299717688429