| L(s) = 1 | − 2-s + 3-s + 4-s + 5-s − 6-s − 2·7-s − 8-s + 9-s − 10-s + 2·11-s + 12-s − 2·13-s + 2·14-s + 15-s + 16-s + 4·17-s − 18-s + 2·19-s + 20-s − 2·21-s − 2·22-s + 2·23-s − 24-s + 25-s + 2·26-s + 27-s − 2·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s − 0.755·7-s − 0.353·8-s + 1/3·9-s − 0.316·10-s + 0.603·11-s + 0.288·12-s − 0.554·13-s + 0.534·14-s + 0.258·15-s + 1/4·16-s + 0.970·17-s − 0.235·18-s + 0.458·19-s + 0.223·20-s − 0.436·21-s − 0.426·22-s + 0.417·23-s − 0.204·24-s + 1/5·25-s + 0.392·26-s + 0.192·27-s − 0.377·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 55470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 55470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.890171885\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.890171885\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 43 | \( 1 \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 - 10 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−14.37215617436809, −14.02376052606345, −13.42384969656538, −12.86050226422398, −12.28932762855111, −11.93086975602964, −11.36775355872882, −10.47828427293212, −10.11048602537225, −9.842412005424023, −9.139391651082041, −8.897799543787285, −8.203376855201276, −7.627351055630225, −7.075948198191572, −6.657724707078113, −5.960387695410445, −5.512165397423131, −4.607544615036610, −4.020576272483873, −3.127166148841054, −2.825816914200990, −2.149866903948840, −1.168800482402771, −0.7307873646742565,
0.7307873646742565, 1.168800482402771, 2.149866903948840, 2.825816914200990, 3.127166148841054, 4.020576272483873, 4.607544615036610, 5.512165397423131, 5.960387695410445, 6.657724707078113, 7.075948198191572, 7.627351055630225, 8.203376855201276, 8.897799543787285, 9.139391651082041, 9.842412005424023, 10.11048602537225, 10.47828427293212, 11.36775355872882, 11.93086975602964, 12.28932762855111, 12.86050226422398, 13.42384969656538, 14.02376052606345, 14.37215617436809
