L(s) = 1 | − 2-s + (−0.230 + 1.71i)3-s + 4-s + (−0.286 + 2.21i)5-s + (0.230 − 1.71i)6-s − 4.09i·7-s − 8-s + (−2.89 − 0.791i)9-s + (0.286 − 2.21i)10-s + 2.91·11-s + (−0.230 + 1.71i)12-s + 2.84·13-s + 4.09i·14-s + (−3.74 − 1.00i)15-s + 16-s − 3.40i·17-s + ⋯ |
L(s) = 1 | − 0.707·2-s + (−0.133 + 0.991i)3-s + 0.5·4-s + (−0.128 + 0.991i)5-s + (0.0941 − 0.700i)6-s − 1.54i·7-s − 0.353·8-s + (−0.964 − 0.263i)9-s + (0.0906 − 0.701i)10-s + 0.879·11-s + (−0.0665 + 0.495i)12-s + 0.790·13-s + 1.09i·14-s + (−0.965 − 0.258i)15-s + 0.250·16-s − 0.826i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.936 - 0.349i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.936 - 0.349i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.06886 + 0.193114i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.06886 + 0.193114i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 + (0.230 - 1.71i)T \) |
| 5 | \( 1 + (0.286 - 2.21i)T \) |
| 31 | \( 1 + (-4.53 + 3.23i)T \) |
good | 7 | \( 1 + 4.09iT - 7T^{2} \) |
| 11 | \( 1 - 2.91T + 11T^{2} \) |
| 13 | \( 1 - 2.84T + 13T^{2} \) |
| 17 | \( 1 + 3.40iT - 17T^{2} \) |
| 19 | \( 1 - 2.22T + 19T^{2} \) |
| 23 | \( 1 + 3.74iT - 23T^{2} \) |
| 29 | \( 1 - 5.67T + 29T^{2} \) |
| 37 | \( 1 + 1.51T + 37T^{2} \) |
| 41 | \( 1 + 0.438iT - 41T^{2} \) |
| 43 | \( 1 - 8.29T + 43T^{2} \) |
| 47 | \( 1 + 7.81T + 47T^{2} \) |
| 53 | \( 1 - 10.9iT - 53T^{2} \) |
| 59 | \( 1 + 2.26iT - 59T^{2} \) |
| 61 | \( 1 + 3.00iT - 61T^{2} \) |
| 67 | \( 1 - 9.68iT - 67T^{2} \) |
| 71 | \( 1 - 11.2iT - 71T^{2} \) |
| 73 | \( 1 - 2.62T + 73T^{2} \) |
| 79 | \( 1 + 15.1iT - 79T^{2} \) |
| 83 | \( 1 - 8.73iT - 83T^{2} \) |
| 89 | \( 1 - 13.2T + 89T^{2} \) |
| 97 | \( 1 - 3.32iT - 97T^{2} \) |
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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
−10.14399356718000379148829122810, −9.544626280579902772155868959258, −8.546386558455283122438405455675, −7.58777095280475839778139446740, −6.76006348745639213390844909942, −6.09299411014723615652975843832, −4.53577470255536523469572645871, −3.76432759935484559013694968696, −2.84331097644270054226744115771, −0.836098398904160244163182128834,
1.12265851557532272583680518858, 2.00815502940552715508706285633, 3.37690208227471016501514541384, 5.03620541847324555311700994594, 5.96539437140837434313816210474, 6.48875969429138337877771935389, 7.77960589614852968898705224200, 8.514083765202961143746998606522, 8.867286945734831890723274347566, 9.702445423994641554230353509060