L(s) = 1 | + i·2-s − 1.78·3-s − 4-s − 1.78i·6-s + 3.14·7-s − i·8-s + 0.191·9-s − 0.908·11-s + 1.78·12-s + 2.22i·13-s + 3.14i·14-s + 16-s − 2.10i·17-s + 0.191i·18-s + 4.16i·19-s + ⋯ |
L(s) = 1 | + 0.707i·2-s − 1.03·3-s − 0.5·4-s − 0.729i·6-s + 1.19·7-s − 0.353i·8-s + 0.0638·9-s − 0.274·11-s + 0.515·12-s + 0.616i·13-s + 0.841i·14-s + 0.250·16-s − 0.509i·17-s + 0.0451i·18-s + 0.955i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.180 - 0.983i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.180 - 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.117988875\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.117988875\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-5.98 - 1.10i)T \) |
good | 3 | \( 1 + 1.78T + 3T^{2} \) |
| 7 | \( 1 - 3.14T + 7T^{2} \) |
| 11 | \( 1 + 0.908T + 11T^{2} \) |
| 13 | \( 1 - 2.22iT - 13T^{2} \) |
| 17 | \( 1 + 2.10iT - 17T^{2} \) |
| 19 | \( 1 - 4.16iT - 19T^{2} \) |
| 23 | \( 1 + 7.66iT - 23T^{2} \) |
| 29 | \( 1 + 2.69iT - 29T^{2} \) |
| 31 | \( 1 - 5.96iT - 31T^{2} \) |
| 41 | \( 1 - 2.32T + 41T^{2} \) |
| 43 | \( 1 + 5.72iT - 43T^{2} \) |
| 47 | \( 1 + 8.89T + 47T^{2} \) |
| 53 | \( 1 - 9.37T + 53T^{2} \) |
| 59 | \( 1 - 5.55iT - 59T^{2} \) |
| 61 | \( 1 - 3.16iT - 61T^{2} \) |
| 67 | \( 1 - 7.64T + 67T^{2} \) |
| 71 | \( 1 - 13.5T + 71T^{2} \) |
| 73 | \( 1 - 2.14T + 73T^{2} \) |
| 79 | \( 1 - 3.35iT - 79T^{2} \) |
| 83 | \( 1 - 16.2T + 83T^{2} \) |
| 89 | \( 1 - 8.35iT - 89T^{2} \) |
| 97 | \( 1 - 5.14iT - 97T^{2} \) |
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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
−9.287783232556759325353721453066, −8.379004217246320234768253459219, −7.920077340015483498500763331940, −6.83683016709868655477989127913, −6.27906862407865928768360790523, −5.29562589216130116489390772614, −4.88514975899101439279762439727, −3.97352578042919934351933256128, −2.35952437458870345060766743344, −0.902365098451485062900719993233,
0.66195650389792710045900698589, 1.83496395446087431319784346412, 3.02670033195858334089310797697, 4.21548925724925889436316796775, 5.12188776145105156784129258389, 5.51155573287086273415316971564, 6.52763057364499427098107733826, 7.72159254639401434841674236560, 8.203936383178404760615196828422, 9.241982515165820151674755528352