| L(s) = 1 | + i·2-s + 2.27i·3-s − 4-s − 2.27·6-s − 1.27i·7-s − i·8-s − 2.16·9-s − 4.43·11-s − 2.27i·12-s − 5.27i·13-s + 1.27·14-s + 16-s + 0.273i·17-s − 2.16i·18-s + 5.71·19-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s + 1.31i·3-s − 0.5·4-s − 0.927·6-s − 0.481i·7-s − 0.353i·8-s − 0.722·9-s − 1.33·11-s − 0.656i·12-s − 1.46i·13-s + 0.340·14-s + 0.250·16-s + 0.0662i·17-s − 0.510i·18-s + 1.31·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.212338870\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.212338870\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - iT \) |
| 5 | \( 1 \) |
| 37 | \( 1 - iT \) |
| good | 3 | \( 1 - 2.27iT - 3T^{2} \) |
| 7 | \( 1 + 1.27iT - 7T^{2} \) |
| 11 | \( 1 + 4.43T + 11T^{2} \) |
| 13 | \( 1 + 5.27iT - 13T^{2} \) |
| 17 | \( 1 - 0.273iT - 17T^{2} \) |
| 19 | \( 1 - 5.71T + 19T^{2} \) |
| 23 | \( 1 + 6.54iT - 23T^{2} \) |
| 29 | \( 1 + 0.546T + 29T^{2} \) |
| 31 | \( 1 + 5.98T + 31T^{2} \) |
| 41 | \( 1 - 11.8T + 41T^{2} \) |
| 43 | \( 1 + 10.3iT - 43T^{2} \) |
| 47 | \( 1 - 47T^{2} \) |
| 53 | \( 1 - 2.33iT - 53T^{2} \) |
| 59 | \( 1 + 2.37T + 59T^{2} \) |
| 61 | \( 1 - 11.8T + 61T^{2} \) |
| 67 | \( 1 + 7.15iT - 67T^{2} \) |
| 71 | \( 1 - 5.60T + 71T^{2} \) |
| 73 | \( 1 + 9.71iT - 73T^{2} \) |
| 79 | \( 1 + 11.4T + 79T^{2} \) |
| 83 | \( 1 - 3.89iT - 83T^{2} \) |
| 89 | \( 1 + 13.8T + 89T^{2} \) |
| 97 | \( 1 + 0.879iT - 97T^{2} \) |
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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
−9.348074024245227565343303025246, −8.492198590487789491008520822320, −7.72143387572546662036446640047, −7.13143207976803616650703169766, −5.68875141951882752413210892787, −5.35958613599392208589860906626, −4.50643348685175503499766205131, −3.61701446792268756661767197073, −2.73951498632441218551425402704, −0.50295495698832067412074754219,
1.15955527483759581664365949180, 2.10435675571612180496112392587, 2.85293382863246096218326658232, 4.08100744676141748616844009783, 5.27745113862383118205432728406, 5.87861624671347683464264242440, 7.08359137800559209096648726563, 7.53961030573544491384366447016, 8.327128807732945091471849870254, 9.323776325912509760907530420174
