L(s) = 1 | + (−3.35 − 1.93i)5-s + (−2.5 − 0.866i)7-s + (3.35 − 1.93i)11-s + 3.46i·13-s + (−2 + 3.46i)19-s + (−6.70 − 3.87i)23-s + (5.00 + 8.66i)25-s + 6.70·29-s + (0.5 + 0.866i)31-s + (6.70 + 7.74i)35-s + (−2 + 3.46i)37-s + 7.74i·41-s + 6.92i·43-s + (−6.70 + 11.6i)47-s + (5.5 + 4.33i)49-s + ⋯ |
L(s) = 1 | + (−1.50 − 0.866i)5-s + (−0.944 − 0.327i)7-s + (1.01 − 0.583i)11-s + 0.960i·13-s + (−0.458 + 0.794i)19-s + (−1.39 − 0.807i)23-s + (1.00 + 1.73i)25-s + 1.24·29-s + (0.0898 + 0.155i)31-s + (1.13 + 1.30i)35-s + (−0.328 + 0.569i)37-s + 1.20i·41-s + 1.05i·43-s + (−0.978 + 1.69i)47-s + (0.785 + 0.618i)49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0633 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0633 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4555432341\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4555432341\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (2.5 + 0.866i)T \) |
good | 5 | \( 1 + (3.35 + 1.93i)T + (2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-3.35 + 1.93i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 - 3.46iT - 13T^{2} \) |
| 17 | \( 1 + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (2 - 3.46i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (6.70 + 3.87i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 - 6.70T + 29T^{2} \) |
| 31 | \( 1 + (-0.5 - 0.866i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (2 - 3.46i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 7.74iT - 41T^{2} \) |
| 43 | \( 1 - 6.92iT - 43T^{2} \) |
| 47 | \( 1 + (6.70 - 11.6i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-3.35 - 5.80i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (3.35 + 5.80i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-9 - 5.19i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-6 + 3.46i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 7.74iT - 71T^{2} \) |
| 73 | \( 1 + (6 - 3.46i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (10.5 + 6.06i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 6.70T + 83T^{2} \) |
| 89 | \( 1 + (6.70 + 3.87i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 5.19iT - 97T^{2} \) |
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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
−10.08273138447760577921052761749, −9.251356532149790697729744734655, −8.433193186912317092813652821737, −7.898251038508377305355842064333, −6.69079304558155988333959294635, −6.17443967569189105077382318803, −4.50561423919173909107225031819, −4.12452520169172423709139938571, −3.16055741595749354742153670064, −1.20503153995458890935588596464,
0.23574733024820085158381789164, 2.46177971153893970063453403682, 3.57417402299498779082517245199, 4.04772475234453435001718670517, 5.48139927121624473964138070826, 6.70728574643668517469699834824, 7.02422873041533328337385869638, 8.072182112075953068794277111720, 8.799663945491098861240055783144, 9.932123542239718294978478186236