L(s) = 1 | + (0.707 − 0.707i)2-s + (−0.931 − 1.46i)3-s − 1.00i·4-s + (2.16 − 0.569i)5-s + (−1.69 − 0.373i)6-s + (0.707 + 0.707i)7-s + (−0.707 − 0.707i)8-s + (−1.26 + 2.72i)9-s + (1.12 − 1.93i)10-s − 6.30i·11-s + (−1.46 + 0.931i)12-s + (−0.977 + 0.977i)13-s + 1.00·14-s + (−2.84 − 2.62i)15-s − 1.00·16-s + (−4.86 + 4.86i)17-s + ⋯ |
L(s) = 1 | + (0.499 − 0.499i)2-s + (−0.537 − 0.843i)3-s − 0.500i·4-s + (0.967 − 0.254i)5-s + (−0.690 − 0.152i)6-s + (0.267 + 0.267i)7-s + (−0.250 − 0.250i)8-s + (−0.421 + 0.906i)9-s + (0.356 − 0.610i)10-s − 1.90i·11-s + (−0.421 + 0.268i)12-s + (−0.271 + 0.271i)13-s + 0.267·14-s + (−0.734 − 0.678i)15-s − 0.250·16-s + (−1.18 + 1.18i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.190 + 0.981i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.190 + 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.934039 - 1.13318i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.934039 - 1.13318i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.707 + 0.707i)T \) |
| 3 | \( 1 + (0.931 + 1.46i)T \) |
| 5 | \( 1 + (-2.16 + 0.569i)T \) |
| 7 | \( 1 + (-0.707 - 0.707i)T \) |
good | 11 | \( 1 + 6.30iT - 11T^{2} \) |
| 13 | \( 1 + (0.977 - 0.977i)T - 13iT^{2} \) |
| 17 | \( 1 + (4.86 - 4.86i)T - 17iT^{2} \) |
| 19 | \( 1 - 0.285iT - 19T^{2} \) |
| 23 | \( 1 + (-4.26 - 4.26i)T + 23iT^{2} \) |
| 29 | \( 1 - 3.84T + 29T^{2} \) |
| 31 | \( 1 - 6.64T + 31T^{2} \) |
| 37 | \( 1 + (-0.317 - 0.317i)T + 37iT^{2} \) |
| 41 | \( 1 - 4.55iT - 41T^{2} \) |
| 43 | \( 1 + (-2.07 + 2.07i)T - 43iT^{2} \) |
| 47 | \( 1 + (6.69 - 6.69i)T - 47iT^{2} \) |
| 53 | \( 1 + (-3.12 - 3.12i)T + 53iT^{2} \) |
| 59 | \( 1 + 13.0T + 59T^{2} \) |
| 61 | \( 1 - 1.09T + 61T^{2} \) |
| 67 | \( 1 + (5.63 + 5.63i)T + 67iT^{2} \) |
| 71 | \( 1 + 5.42iT - 71T^{2} \) |
| 73 | \( 1 + (3.69 - 3.69i)T - 73iT^{2} \) |
| 79 | \( 1 - 4.38iT - 79T^{2} \) |
| 83 | \( 1 + (1.52 + 1.52i)T + 83iT^{2} \) |
| 89 | \( 1 - 8.96T + 89T^{2} \) |
| 97 | \( 1 + (-1.50 - 1.50i)T + 97iT^{2} \) |
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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
−12.14426284136010829798107251805, −11.19755914443981772702259898634, −10.59644290331386273212066370255, −9.104034498194567954963372168447, −8.206861352831441106593922674313, −6.44470233150105522846585784922, −5.92964108140299805088978615556, −4.80718839586389561963659813570, −2.81389086568044070757467002087, −1.37242599095823206284890245803,
2.61550670295574961636560240702, 4.57346343434114360333979858770, 4.98887735503639322283348798092, 6.46306143082641331420558470941, 7.15732085378312382072242446686, 8.897712998266924308401968178849, 9.818482521613996229322239252646, 10.57042705301724303100850195793, 11.73834532878745229785497547473, 12.68677597736273730523600925385