| L(s) = 1 | − i·2-s − 4-s + 3i·5-s + 2i·7-s + i·8-s + 3·10-s − 6i·11-s + 2·14-s + 16-s − 3·17-s − 2i·19-s − 3i·20-s − 6·22-s − 6·23-s − 4·25-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.5·4-s + 1.34i·5-s + 0.755i·7-s + 0.353i·8-s + 0.948·10-s − 1.80i·11-s + 0.534·14-s + 0.250·16-s − 0.727·17-s − 0.458i·19-s − 0.670i·20-s − 1.27·22-s − 1.25·23-s − 0.800·25-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3042 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.554 + 0.832i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3042 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.554 + 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8540046922\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8540046922\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + iT \) |
| 3 | \( 1 \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 - 3iT - 5T^{2} \) |
| 7 | \( 1 - 2iT - 7T^{2} \) |
| 11 | \( 1 + 6iT - 11T^{2} \) |
| 17 | \( 1 + 3T + 17T^{2} \) |
| 19 | \( 1 + 2iT - 19T^{2} \) |
| 23 | \( 1 + 6T + 23T^{2} \) |
| 29 | \( 1 + 3T + 29T^{2} \) |
| 31 | \( 1 - 4iT - 31T^{2} \) |
| 37 | \( 1 + 7iT - 37T^{2} \) |
| 41 | \( 1 + 3iT - 41T^{2} \) |
| 43 | \( 1 - 10T + 43T^{2} \) |
| 47 | \( 1 + 6iT - 47T^{2} \) |
| 53 | \( 1 + 3T + 53T^{2} \) |
| 59 | \( 1 - 59T^{2} \) |
| 61 | \( 1 + 7T + 61T^{2} \) |
| 67 | \( 1 - 10iT - 67T^{2} \) |
| 71 | \( 1 - 6iT - 71T^{2} \) |
| 73 | \( 1 + 13iT - 73T^{2} \) |
| 79 | \( 1 + 4T + 79T^{2} \) |
| 83 | \( 1 + 6iT - 83T^{2} \) |
| 89 | \( 1 + 18iT - 89T^{2} \) |
| 97 | \( 1 + 14iT - 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.775347656742341972043894041688, −7.77396774518425991021370591890, −6.93129654183147120621234193936, −5.95286256421286728361577274680, −5.67199493111076621280458218394, −4.28109814095154246856433026863, −3.40574834170989541419037801669, −2.78703387214238670327121021428, −1.99678281305320550289416070561, −0.28375944327736613717004271617,
1.15507579099097851913742935470, 2.18425003754090017384578551295, 3.94817747707020312286286318964, 4.42244291550634203537230401192, 4.98881721024624112984027305687, 5.97086564337813197198563685615, 6.75969561148304226372221558972, 7.68638894058128807497550973040, 7.947305135300674233312556026942, 8.994131226139116208460147131087
