L(s) = 1 | + (0.322 − 0.558i)2-s + (−1.69 + 0.358i)3-s + (0.792 + 1.37i)4-s + (1.60 − 1.56i)5-s + (−0.346 + 1.06i)6-s + (2.64 − 0.105i)7-s + 2.31·8-s + (2.74 − 1.21i)9-s + (−0.354 − 1.39i)10-s + (−3.51 + 2.02i)11-s + (−1.83 − 2.04i)12-s − 4.21·13-s + (0.793 − 1.51i)14-s + (−2.15 + 3.21i)15-s + (−0.839 + 1.45i)16-s + (1.88 − 1.08i)17-s + ⋯ |
L(s) = 1 | + (0.227 − 0.394i)2-s + (−0.978 + 0.206i)3-s + (0.396 + 0.685i)4-s + (0.716 − 0.697i)5-s + (−0.141 + 0.433i)6-s + (0.999 − 0.0397i)7-s + 0.817·8-s + (0.914 − 0.404i)9-s + (−0.112 − 0.441i)10-s + (−1.05 + 0.611i)11-s + (−0.529 − 0.589i)12-s − 1.16·13-s + (0.212 − 0.403i)14-s + (−0.556 + 0.830i)15-s + (−0.209 + 0.363i)16-s + (0.457 − 0.263i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.988 + 0.153i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.988 + 0.153i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.07555 - 0.0827929i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.07555 - 0.0827929i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (1.69 - 0.358i)T \) |
| 5 | \( 1 + (-1.60 + 1.56i)T \) |
| 7 | \( 1 + (-2.64 + 0.105i)T \) |
good | 2 | \( 1 + (-0.322 + 0.558i)T + (-1 - 1.73i)T^{2} \) |
| 11 | \( 1 + (3.51 - 2.02i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + 4.21T + 13T^{2} \) |
| 17 | \( 1 + (-1.88 + 1.08i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (3.87 + 2.23i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (0.322 - 0.558i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 1.16iT - 29T^{2} \) |
| 31 | \( 1 + (0.339 - 0.195i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (3.69 + 2.13i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 2.27T + 41T^{2} \) |
| 43 | \( 1 - 6.54iT - 43T^{2} \) |
| 47 | \( 1 + (6.75 + 3.90i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (3.60 + 6.23i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-5.66 - 9.80i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-6.05 - 3.49i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-7.56 + 4.36i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 8.13iT - 71T^{2} \) |
| 73 | \( 1 + (-2.61 - 4.53i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (1.87 - 3.24i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 5.27iT - 83T^{2} \) |
| 89 | \( 1 + (-0.447 + 0.774i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 3.89T + 97T^{2} \) |
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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
−13.30112666289987877665261960788, −12.55350926577213789202778000531, −11.79041592939548977080935088532, −10.71954122160454164731168613384, −9.816572338572298069429840579782, −8.146704305495724404513368627869, −7.01506799851600345410450559935, −5.26220413539681490268758575800, −4.55540681064617519704862514331, −2.11768357566233185544331273952,
2.02337851965601909458889220254, 4.98144371842320715821640481892, 5.69897226240747832319658062036, 6.81844848781462582551565282773, 7.88669344856195060009567825630, 10.06359113296986825621859102373, 10.61112686206751738475219658481, 11.47916677248235808111968510286, 12.81150088257920980260342487524, 14.02160572997561743303882672026