L(s) = 1 | − 2-s + (−1.03 − 1.38i)3-s + 4-s + (1.98 + 1.02i)5-s + (1.03 + 1.38i)6-s − 8-s + (−0.844 + 2.87i)9-s + (−1.98 − 1.02i)10-s + 1.27i·11-s + (−1.03 − 1.38i)12-s + 1.32·13-s + (−0.643 − 3.81i)15-s + 16-s − 5.76i·17-s + (0.844 − 2.87i)18-s + 2.44i·19-s + ⋯ |
L(s) = 1 | − 0.707·2-s + (−0.599 − 0.800i)3-s + 0.5·4-s + (0.888 + 0.457i)5-s + (0.423 + 0.566i)6-s − 0.353·8-s + (−0.281 + 0.959i)9-s + (−0.628 − 0.323i)10-s + 0.385i·11-s + (−0.299 − 0.400i)12-s + 0.366·13-s + (−0.166 − 0.986i)15-s + 0.250·16-s − 1.39i·17-s + (0.199 − 0.678i)18-s + 0.560i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.00978i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.00978i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.126592677\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.126592677\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (1.03 + 1.38i)T \) |
| 5 | \( 1 + (-1.98 - 1.02i)T \) |
| 7 | \( 1 \) |
good | 11 | \( 1 - 1.27iT - 11T^{2} \) |
| 13 | \( 1 - 1.32T + 13T^{2} \) |
| 17 | \( 1 + 5.76iT - 17T^{2} \) |
| 19 | \( 1 - 2.44iT - 19T^{2} \) |
| 23 | \( 1 + 0.680T + 23T^{2} \) |
| 29 | \( 1 + 6.12iT - 29T^{2} \) |
| 31 | \( 1 - 6.94iT - 31T^{2} \) |
| 37 | \( 1 - 4.67iT - 37T^{2} \) |
| 41 | \( 1 - 12.3T + 41T^{2} \) |
| 43 | \( 1 - 1.87iT - 43T^{2} \) |
| 47 | \( 1 - 7.97iT - 47T^{2} \) |
| 53 | \( 1 + 6.98T + 53T^{2} \) |
| 59 | \( 1 - 9.65T + 59T^{2} \) |
| 61 | \( 1 + 4.22iT - 61T^{2} \) |
| 67 | \( 1 - 12.9iT - 67T^{2} \) |
| 71 | \( 1 + 5.09iT - 71T^{2} \) |
| 73 | \( 1 - 16.4T + 73T^{2} \) |
| 79 | \( 1 + 15.5T + 79T^{2} \) |
| 83 | \( 1 + 3.33iT - 83T^{2} \) |
| 89 | \( 1 + 12.7T + 89T^{2} \) |
| 97 | \( 1 - 2.90T + 97T^{2} \) |
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
−9.663101475201918954492413441474, −8.709540733104288476807480529622, −7.73721064030638027964760078007, −7.11795463025417241376149924970, −6.33683130539443849661912147814, −5.72485370406045850584938294199, −4.70849764522369645003502808675, −2.97360295381570568542584461075, −2.10114403533156691246294503647, −1.00988876046762806493764280032,
0.78254567530567599785394652482, 2.11689007524147688179338599379, 3.50071039676040608720315310209, 4.50663239714620709257376223638, 5.65596940212675835309710231254, 6.01396321384290191394839588597, 6.96549532258274026578689631481, 8.242069264232254375749592701298, 8.891877319260579652055100339380, 9.500748295716768128669682824565