L(s) = 1 | + (1.99 + 0.169i)2-s + 1.73i·3-s + (3.94 + 0.675i)4-s + (−0.293 + 3.45i)6-s + 12.3i·7-s + (7.74 + 2.01i)8-s − 2.99·9-s − 11.0i·11-s + (−1.16 + 6.82i)12-s − 2.82·13-s + (−2.10 + 24.7i)14-s + (15.0 + 5.32i)16-s − 6.52·17-s + (−5.97 − 0.508i)18-s + 27.9i·19-s + ⋯ |
L(s) = 1 | + (0.996 + 0.0847i)2-s + 0.577i·3-s + (0.985 + 0.168i)4-s + (−0.0489 + 0.575i)6-s + 1.77i·7-s + (0.967 + 0.251i)8-s − 0.333·9-s − 1.00i·11-s + (−0.0974 + 0.569i)12-s − 0.216·13-s + (−0.150 + 1.76i)14-s + (0.942 + 0.332i)16-s − 0.383·17-s + (−0.332 − 0.0282i)18-s + 1.47i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.168 - 0.985i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.168 - 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(2.27321 + 1.91693i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.27321 + 1.91693i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.99 - 0.169i)T \) |
| 3 | \( 1 - 1.73iT \) |
| 5 | \( 1 \) |
good | 7 | \( 1 - 12.3iT - 49T^{2} \) |
| 11 | \( 1 + 11.0iT - 121T^{2} \) |
| 13 | \( 1 + 2.82T + 169T^{2} \) |
| 17 | \( 1 + 6.52T + 289T^{2} \) |
| 19 | \( 1 - 27.9iT - 361T^{2} \) |
| 23 | \( 1 + 7.90iT - 529T^{2} \) |
| 29 | \( 1 - 50.7T + 841T^{2} \) |
| 31 | \( 1 + 36.3iT - 961T^{2} \) |
| 37 | \( 1 - 18.9T + 1.36e3T^{2} \) |
| 41 | \( 1 - 5.30T + 1.68e3T^{2} \) |
| 43 | \( 1 + 45.5iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 11.7iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 41.1T + 2.80e3T^{2} \) |
| 59 | \( 1 + 10.7iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 56.1T + 3.72e3T^{2} \) |
| 67 | \( 1 + 16.1iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 66.1iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 15.6T + 5.32e3T^{2} \) |
| 79 | \( 1 + 123. iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 99.6iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 101.T + 7.92e3T^{2} \) |
| 97 | \( 1 + 127.T + 9.40e3T^{2} \) |
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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
−11.91638480428719063258808068363, −11.06338011746953277783516918635, −9.992725978226262075728399974994, −8.771943706499456742695856333735, −8.009581653218328742810014986709, −6.25368428060902073877598584942, −5.76075754567050115038566140064, −4.69244066526921682013663499190, −3.34667578868649304567467528669, −2.29731079493737612065341272348,
1.15391622641662921526962140105, 2.76052223612923344602076858706, 4.20537693342380536323840486745, 4.96220299339527526628964254346, 6.75442940324712039315169348314, 6.95543456679686346618312360136, 8.026127869561781486906620284565, 9.743158764543393729036676935371, 10.64394931566110909266269473657, 11.40267776648102667285962901210