| L(s) = 1 | + (2.03 + 0.741i)2-s + (0.538 + 0.451i)4-s + (4.99 − 0.211i)5-s + (2.32 + 2.76i)7-s + (−3.57 − 6.19i)8-s + (10.3 + 3.27i)10-s + (8.70 − 1.53i)11-s + (3.15 + 8.66i)13-s + (2.68 + 7.36i)14-s + (−3.18 − 18.0i)16-s + (9.89 − 17.1i)17-s + (12.1 + 21.0i)19-s + (2.78 + 2.14i)20-s + (18.8 + 3.32i)22-s + (−4.74 − 3.98i)23-s + ⋯ |
| L(s) = 1 | + (1.01 + 0.370i)2-s + (0.134 + 0.112i)4-s + (0.999 − 0.0423i)5-s + (0.332 + 0.395i)7-s + (−0.446 − 0.773i)8-s + (1.03 + 0.327i)10-s + (0.790 − 0.139i)11-s + (0.242 + 0.666i)13-s + (0.191 + 0.526i)14-s + (−0.198 − 1.12i)16-s + (0.581 − 1.00i)17-s + (0.640 + 1.10i)19-s + (0.139 + 0.107i)20-s + (0.857 + 0.151i)22-s + (−0.206 − 0.173i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.970 - 0.241i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.970 - 0.241i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(3.36874 + 0.412019i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.36874 + 0.412019i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + (-4.99 + 0.211i)T \) |
| good | 2 | \( 1 + (-2.03 - 0.741i)T + (3.06 + 2.57i)T^{2} \) |
| 7 | \( 1 + (-2.32 - 2.76i)T + (-8.50 + 48.2i)T^{2} \) |
| 11 | \( 1 + (-8.70 + 1.53i)T + (113. - 41.3i)T^{2} \) |
| 13 | \( 1 + (-3.15 - 8.66i)T + (-129. + 108. i)T^{2} \) |
| 17 | \( 1 + (-9.89 + 17.1i)T + (-144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (-12.1 - 21.0i)T + (-180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (4.74 + 3.98i)T + (91.8 + 520. i)T^{2} \) |
| 29 | \( 1 + (-5.21 + 14.3i)T + (-644. - 540. i)T^{2} \) |
| 31 | \( 1 + (9.90 + 8.30i)T + (166. + 946. i)T^{2} \) |
| 37 | \( 1 + (-9.23 - 5.33i)T + (684.5 + 1.18e3i)T^{2} \) |
| 41 | \( 1 + (-23.8 - 65.6i)T + (-1.28e3 + 1.08e3i)T^{2} \) |
| 43 | \( 1 + (58.3 - 10.2i)T + (1.73e3 - 632. i)T^{2} \) |
| 47 | \( 1 + (12.6 - 10.6i)T + (383. - 2.17e3i)T^{2} \) |
| 53 | \( 1 + 85.3T + 2.80e3T^{2} \) |
| 59 | \( 1 + (-60.9 - 10.7i)T + (3.27e3 + 1.19e3i)T^{2} \) |
| 61 | \( 1 + (-55.8 + 46.8i)T + (646. - 3.66e3i)T^{2} \) |
| 67 | \( 1 + (7.87 + 21.6i)T + (-3.43e3 + 2.88e3i)T^{2} \) |
| 71 | \( 1 + (49.0 + 28.3i)T + (2.52e3 + 4.36e3i)T^{2} \) |
| 73 | \( 1 + (101. - 58.5i)T + (2.66e3 - 4.61e3i)T^{2} \) |
| 79 | \( 1 + (7.55 + 2.74i)T + (4.78e3 + 4.01e3i)T^{2} \) |
| 83 | \( 1 + (77.8 + 28.3i)T + (5.27e3 + 4.42e3i)T^{2} \) |
| 89 | \( 1 + (102. - 59.2i)T + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 + (30.2 - 5.33i)T + (8.84e3 - 3.21e3i)T^{2} \) |
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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
−11.41022904413930782088351489577, −9.789509180492964998170355376160, −9.564142847908740266365034461241, −8.318189021576787232705172968844, −6.89761795372267287319800413983, −6.08332062240584244810943013089, −5.35554792252458517537170815786, −4.36402877061039398622096430582, −3.08063057319770517808249212208, −1.43386811112935084723178372480,
1.50733841780876894006455773695, 2.93755085747086197982313934307, 4.00655434821032055108904168621, 5.15206062168620223805988900945, 5.88224651207193281743476407105, 7.02773738855372800325314695480, 8.360892370711151785481712432287, 9.231565584643210445223213478334, 10.30801486639637551194076786495, 11.11643738273245145661500596661
