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
\(L(1)\) |
\(\approx\) |
\(0.8522641906\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8522641906\) |
\(L(\frac{3}{2})\) |
|
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
−9.025754220021000174979437096212, −8.389762371551323913008078448412, −7.45393131460687597636666657406, −6.78393093679093545719787186567, −6.09078602442660415306002097718, −4.99850548826578980713085435448, −4.02564724950893692762776857542, −3.09655248096902911698607295815, −2.32866448519973008926440519587, −0.25959341148870349575750618633,
1.54708412981970110047044331518, 2.75505739417521699564818720734, 3.33472864266056226324591875078, 3.98928618403321034078562666891, 5.41120614644119611688352026684, 6.09198964222807726224889512010, 7.31615014338280687871938400453, 8.032060851131664395920337624997, 8.762452124173542440400309832542, 9.499251050765775831181314278624