| L(s) = 1 | − 2-s + (−0.423 + 1.67i)3-s + 4-s − 4.07i·5-s + (0.423 − 1.67i)6-s − 4.79i·7-s − 8-s + (−2.64 − 1.42i)9-s + 4.07i·10-s + (2.78 + 1.80i)11-s + (−0.423 + 1.67i)12-s − 2.57i·13-s + 4.79i·14-s + (6.83 + 1.72i)15-s + 16-s + 17-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + (−0.244 + 0.969i)3-s + 0.5·4-s − 1.82i·5-s + (0.172 − 0.685i)6-s − 1.81i·7-s − 0.353·8-s + (−0.880 − 0.474i)9-s + 1.28i·10-s + (0.838 + 0.544i)11-s + (−0.122 + 0.484i)12-s − 0.713i·13-s + 1.28i·14-s + (1.76 + 0.445i)15-s + 0.250·16-s + 0.242·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1122 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.733 + 0.680i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1122 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.733 + 0.680i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7559620445\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7559620445\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (0.423 - 1.67i)T \) |
| 11 | \( 1 + (-2.78 - 1.80i)T \) |
| 17 | \( 1 - T \) |
| good | 5 | \( 1 + 4.07iT - 5T^{2} \) |
| 7 | \( 1 + 4.79iT - 7T^{2} \) |
| 13 | \( 1 + 2.57iT - 13T^{2} \) |
| 19 | \( 1 + 4.54iT - 19T^{2} \) |
| 23 | \( 1 - 5.62iT - 23T^{2} \) |
| 29 | \( 1 + 7.91T + 29T^{2} \) |
| 31 | \( 1 + 0.355T + 31T^{2} \) |
| 37 | \( 1 - 5.35T + 37T^{2} \) |
| 41 | \( 1 + 1.52T + 41T^{2} \) |
| 43 | \( 1 + 4.45iT - 43T^{2} \) |
| 47 | \( 1 - 9.23iT - 47T^{2} \) |
| 53 | \( 1 - 9.72iT - 53T^{2} \) |
| 59 | \( 1 + 11.6iT - 59T^{2} \) |
| 61 | \( 1 - 4.09iT - 61T^{2} \) |
| 67 | \( 1 + 5.11T + 67T^{2} \) |
| 71 | \( 1 + 7.91iT - 71T^{2} \) |
| 73 | \( 1 + 4.26iT - 73T^{2} \) |
| 79 | \( 1 - 9.05iT - 79T^{2} \) |
| 83 | \( 1 - 7.26T + 83T^{2} \) |
| 89 | \( 1 + 12.2iT - 89T^{2} \) |
| 97 | \( 1 - 0.0605T + 97T^{2} \) |
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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
−9.412700092277120658445173672524, −9.075063015992776623342356309023, −7.929276477777777032349390083809, −7.33925503261369224503242844712, −6.03404511508028251294740805711, −5.01816808103836309478100511171, −4.31377991204197309142202123117, −3.54762232565277000969620300974, −1.39033026975713530921000081538, −0.45028484686944589936261141845,
1.87716455257052517110392633670, 2.52940785521635738571472734410, 3.51752488119468549981051049764, 5.62901604201790531628366890727, 6.23250601270323909463762774860, 6.70239392278305476153878511554, 7.64570788114191744568104605202, 8.463248277506496414120271695983, 9.204892328142710567583040532837, 10.15367159366157375095066293071
