| L(s) = 1 | − 2.27i·2-s − 0.632i·3-s − 3.16·4-s − 1.43·6-s − 3.43i·7-s + 2.64i·8-s + 2.60·9-s − 3.10·11-s + 2i·12-s + 0.563i·13-s − 7.80·14-s − 0.317·16-s + 1.74i·17-s − 5.90i·18-s − 4.53·19-s + ⋯ |
| L(s) = 1 | − 1.60i·2-s − 0.364i·3-s − 1.58·4-s − 0.586·6-s − 1.29i·7-s + 0.935i·8-s + 0.866·9-s − 0.937·11-s + 0.577i·12-s + 0.156i·13-s − 2.08·14-s − 0.0792·16-s + 0.422i·17-s − 1.39i·18-s − 1.04·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 775 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 775 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.553165 + 0.895041i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.553165 + 0.895041i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 31 | \( 1 - T \) |
| good | 2 | \( 1 + 2.27iT - 2T^{2} \) |
| 3 | \( 1 + 0.632iT - 3T^{2} \) |
| 7 | \( 1 + 3.43iT - 7T^{2} \) |
| 11 | \( 1 + 3.10T + 11T^{2} \) |
| 13 | \( 1 - 0.563iT - 13T^{2} \) |
| 17 | \( 1 - 1.74iT - 17T^{2} \) |
| 19 | \( 1 + 4.53T + 19T^{2} \) |
| 23 | \( 1 + 9.24iT - 23T^{2} \) |
| 29 | \( 1 + 4.17T + 29T^{2} \) |
| 37 | \( 1 - 0.804iT - 37T^{2} \) |
| 41 | \( 1 - 9.97T + 41T^{2} \) |
| 43 | \( 1 - 3.74iT - 43T^{2} \) |
| 47 | \( 1 - 2.73iT - 47T^{2} \) |
| 53 | \( 1 - 10.7iT - 53T^{2} \) |
| 59 | \( 1 + 8.90T + 59T^{2} \) |
| 61 | \( 1 - 4.73T + 61T^{2} \) |
| 67 | \( 1 - 0.891iT - 67T^{2} \) |
| 71 | \( 1 + 3.60T + 71T^{2} \) |
| 73 | \( 1 + 8.98iT - 73T^{2} \) |
| 79 | \( 1 - 14.3T + 79T^{2} \) |
| 83 | \( 1 + 14.9iT - 83T^{2} \) |
| 89 | \( 1 + 10.5T + 89T^{2} \) |
| 97 | \( 1 + 9.80iT - 97T^{2} \) |
| 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
−10.21414404196266817930312545660, −9.214453577734760495404011732542, −8.110313495497696628814127578633, −7.25722555441081836619259567566, −6.27213523417891378839643060129, −4.46662093874874642604634846965, −4.23331596998049012753574202516, −2.84517406888503430168437077492, −1.76649139901353663313021121708, −0.52674049277872818768373984308,
2.28419893284315979396043093260, 3.89275328375406583317776143891, 5.13503849917505874432381951020, 5.50331151243891432136769893450, 6.51536449280952127216742581534, 7.48504466577483373303931614132, 8.093439474040232734978682002005, 9.102492016754367962938726900150, 9.572010759947071507445982772418, 10.71808775804555272228475791201
