L(s) = 1 | − 3i·3-s + 13i·7-s − 9·9-s − 23i·13-s + 11·19-s + 39·21-s + 27i·27-s − 59·31-s + 26i·37-s − 69·39-s + 83i·43-s − 120·49-s − 33i·57-s − 121·61-s − 117i·63-s + ⋯ |
L(s) = 1 | − i·3-s + 1.85i·7-s − 9-s − 1.76i·13-s + 0.578·19-s + 1.85·21-s + i·27-s − 1.90·31-s + 0.702i·37-s − 1.76·39-s + 1.93i·43-s − 2.44·49-s − 0.578i·57-s − 1.98·61-s − 1.85i·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.5414333018\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5414333018\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + 3iT \) |
| 5 | \( 1 \) |
good | 7 | \( 1 - 13iT - 49T^{2} \) |
| 11 | \( 1 - 121T^{2} \) |
| 13 | \( 1 + 23iT - 169T^{2} \) |
| 17 | \( 1 + 289T^{2} \) |
| 19 | \( 1 - 11T + 361T^{2} \) |
| 23 | \( 1 + 529T^{2} \) |
| 29 | \( 1 - 841T^{2} \) |
| 31 | \( 1 + 59T + 961T^{2} \) |
| 37 | \( 1 - 26iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 1.68e3T^{2} \) |
| 43 | \( 1 - 83iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 2.20e3T^{2} \) |
| 53 | \( 1 + 2.80e3T^{2} \) |
| 59 | \( 1 - 3.48e3T^{2} \) |
| 61 | \( 1 + 121T + 3.72e3T^{2} \) |
| 67 | \( 1 - 13iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 5.04e3T^{2} \) |
| 73 | \( 1 - 46iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 142T + 6.24e3T^{2} \) |
| 83 | \( 1 + 6.88e3T^{2} \) |
| 89 | \( 1 - 7.92e3T^{2} \) |
| 97 | \( 1 - 167iT - 9.40e3T^{2} \) |
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
−9.596125079566744108661686525046, −8.881024066759561667128452120861, −8.114061847201436409623374491730, −7.54488443915516510283392574198, −6.32804281240571736527096397442, −5.66828971538788621540513019141, −5.13650467608733987479659882388, −3.19981813948027966126231581276, −2.62016294013058508604226690154, −1.42332371151406899348358766287,
0.15677652851306412910321358092, 1.74377695506693417430781020040, 3.41512080014130764067236388147, 4.07092034965117253693955869920, 4.69938630315114602413932672281, 5.83606009752520577561577590139, 7.00816169936187185627896678697, 7.44603916369347235577420959500, 8.728600786314345023625280564125, 9.389777395671716201513469034411