L(s) = 1 | − i·2-s + 2.93i·3-s − 4-s + 2.93·6-s + 1.31i·7-s + i·8-s − 5.63·9-s − 0.258·11-s − 2.93i·12-s − 5.87i·13-s + 1.31·14-s + 16-s − 4.25i·17-s + 5.63i·18-s − 2.93·19-s + ⋯ |
L(s) = 1 | − 0.707i·2-s + 1.69i·3-s − 0.5·4-s + 1.19·6-s + 0.498i·7-s + 0.353i·8-s − 1.87·9-s − 0.0780·11-s − 0.848i·12-s − 1.63i·13-s + 0.352·14-s + 0.250·16-s − 1.03i·17-s + 1.32i·18-s − 0.674·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.027114105\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.027114105\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - 2.93iT - 3T^{2} \) |
| 7 | \( 1 - 1.31iT - 7T^{2} \) |
| 11 | \( 1 + 0.258T + 11T^{2} \) |
| 13 | \( 1 + 5.87iT - 13T^{2} \) |
| 17 | \( 1 + 4.25iT - 17T^{2} \) |
| 19 | \( 1 + 2.93T + 19T^{2} \) |
| 23 | \( 1 + 8.51iT - 23T^{2} \) |
| 29 | \( 1 - 3.61T + 29T^{2} \) |
| 31 | \( 1 - 3.95T + 31T^{2} \) |
| 41 | \( 1 + 8.89T + 41T^{2} \) |
| 43 | \( 1 + 5.61iT - 43T^{2} \) |
| 47 | \( 1 + 5.57iT - 47T^{2} \) |
| 53 | \( 1 - 12.8iT - 53T^{2} \) |
| 59 | \( 1 + 9.57T + 59T^{2} \) |
| 61 | \( 1 - 0.380T + 61T^{2} \) |
| 67 | \( 1 + 5.57iT - 67T^{2} \) |
| 71 | \( 1 - 11.2T + 71T^{2} \) |
| 73 | \( 1 - 2.51iT - 73T^{2} \) |
| 79 | \( 1 - 7.69T + 79T^{2} \) |
| 83 | \( 1 + 6.17iT - 83T^{2} \) |
| 89 | \( 1 + 6T + 89T^{2} \) |
| 97 | \( 1 - 12.8iT - 97T^{2} \) |
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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
−9.206486288960114248316082748412, −8.653074672321881069027119979604, −7.980705559569899381085047525003, −6.47979216737316070072975975922, −5.42589050569464339480209925375, −4.91200565730947830165633826937, −4.12683996425160098723805697133, −3.07955855272880955105606760307, −2.57122562695798069984743080266, −0.40337688963685635378828145153,
1.25094023644387748916338177615, 2.06852849592305826783907407138, 3.54462999319106769299832481617, 4.58176619136600413628180498022, 5.73611461222423198673344552840, 6.58953045969076808084339544304, 6.81376343578509923034407960338, 7.75242661551873148106796989082, 8.255858440256632015554760890426, 9.065175742692582635505679164399