L(s) = 1 | + (1.20 − 0.746i)2-s + (2.18 + 2.18i)3-s + (0.886 − 1.79i)4-s + (−0.707 + 0.707i)5-s + (4.26 + 0.996i)6-s − i·7-s + (−0.272 − 2.81i)8-s + 6.57i·9-s + (−0.321 + 1.37i)10-s + (0.847 − 0.847i)11-s + (5.86 − 1.98i)12-s + (3.07 + 3.07i)13-s + (−0.746 − 1.20i)14-s − 3.09·15-s + (−2.42 − 3.17i)16-s − 5.75·17-s + ⋯ |
L(s) = 1 | + (0.849 − 0.527i)2-s + (1.26 + 1.26i)3-s + (0.443 − 0.896i)4-s + (−0.316 + 0.316i)5-s + (1.73 + 0.406i)6-s − 0.377i·7-s + (−0.0963 − 0.995i)8-s + 2.19i·9-s + (−0.101 + 0.435i)10-s + (0.255 − 0.255i)11-s + (1.69 − 0.572i)12-s + (0.853 + 0.853i)13-s + (−0.199 − 0.321i)14-s − 0.799·15-s + (−0.606 − 0.794i)16-s − 1.39·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.966 - 0.256i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.966 - 0.256i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.25578 + 0.424934i\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.25578 + 0.424934i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.20 + 0.746i)T \) |
| 5 | \( 1 + (0.707 - 0.707i)T \) |
| 7 | \( 1 + iT \) |
good | 3 | \( 1 + (-2.18 - 2.18i)T + 3iT^{2} \) |
| 11 | \( 1 + (-0.847 + 0.847i)T - 11iT^{2} \) |
| 13 | \( 1 + (-3.07 - 3.07i)T + 13iT^{2} \) |
| 17 | \( 1 + 5.75T + 17T^{2} \) |
| 19 | \( 1 + (-3.49 - 3.49i)T + 19iT^{2} \) |
| 23 | \( 1 + 8.84iT - 23T^{2} \) |
| 29 | \( 1 + (2.20 + 2.20i)T + 29iT^{2} \) |
| 31 | \( 1 + 5.44T + 31T^{2} \) |
| 37 | \( 1 + (-4.16 + 4.16i)T - 37iT^{2} \) |
| 41 | \( 1 + 4.94iT - 41T^{2} \) |
| 43 | \( 1 + (7.34 - 7.34i)T - 43iT^{2} \) |
| 47 | \( 1 - 5.42T + 47T^{2} \) |
| 53 | \( 1 + (2.14 - 2.14i)T - 53iT^{2} \) |
| 59 | \( 1 + (3.49 - 3.49i)T - 59iT^{2} \) |
| 61 | \( 1 + (3.45 + 3.45i)T + 61iT^{2} \) |
| 67 | \( 1 + (-3.12 - 3.12i)T + 67iT^{2} \) |
| 71 | \( 1 + 0.245iT - 71T^{2} \) |
| 73 | \( 1 + 0.160iT - 73T^{2} \) |
| 79 | \( 1 + 9.95T + 79T^{2} \) |
| 83 | \( 1 + (-6.68 - 6.68i)T + 83iT^{2} \) |
| 89 | \( 1 + 6.01iT - 89T^{2} \) |
| 97 | \( 1 + 14.8T + 97T^{2} \) |
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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
−10.91067193406271187478580707423, −10.02140034866195760492784121797, −9.190924958316848194008346599412, −8.452937025101130353911051002170, −7.17220259824647487122930954705, −6.03890443416529570993200213526, −4.54315077638393187969875040419, −4.07113261216965835807880662885, −3.23261102851108217822365725593, −2.09222595480847040113179946672,
1.69950461851462465215820137114, 2.96709409372212028280182478488, 3.76447370037400765767505027542, 5.23853429119627532371470704900, 6.36474110072541201015494068611, 7.23200114515084372146954226823, 7.84450442158538384650287071708, 8.714798993169498982658747483797, 9.275021145954656657064145838627, 11.21072004998058879014659644854