L(s) = 1 | + (−0.866 + 0.5i)3-s + (0.5 + 0.866i)4-s + (0.866 − 0.5i)7-s + (0.499 − 0.866i)9-s + (−0.866 − 0.499i)12-s + i·13-s + (−0.499 + 0.866i)16-s + (−0.5 + 0.866i)19-s + (−0.499 + 0.866i)21-s + 0.999i·27-s + (0.866 + 0.499i)28-s + (−1 − 1.73i)31-s + 0.999·36-s + (0.866 + 0.5i)37-s + (−0.5 − 0.866i)39-s + ⋯ |
L(s) = 1 | + (−0.866 + 0.5i)3-s + (0.5 + 0.866i)4-s + (0.866 − 0.5i)7-s + (0.499 − 0.866i)9-s + (−0.866 − 0.499i)12-s + i·13-s + (−0.499 + 0.866i)16-s + (−0.5 + 0.866i)19-s + (−0.499 + 0.866i)21-s + 0.999i·27-s + (0.866 + 0.499i)28-s + (−1 − 1.73i)31-s + 0.999·36-s + (0.866 + 0.5i)37-s + (−0.5 − 0.866i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.497 - 0.867i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.497 - 0.867i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8144140001\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8144140001\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.866 - 0.5i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-0.866 + 0.5i)T \) |
good | 2 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 - iT - T^{2} \) |
| 17 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 19 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 + 2iT - T^{2} \) |
| 47 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 53 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 59 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 79 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 + T^{2} \) |
| 89 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 97 | \( 1 + iT - T^{2} \) |
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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
−11.29086795088706991558262623613, −10.61662149592261850202692796156, −9.563383376064089929864771658994, −8.485526709763326895231228389587, −7.52305118607046320499038417515, −6.71585031807982118020629669133, −5.67969635074650773342840217999, −4.37968090057031358885415286030, −3.80054616729598062928015936041, −1.94263208975749447212563756240,
1.31700970631783550491257665869, 2.57763652978907088229204792629, 4.72499988851361415917537063770, 5.40749832722790060450139866586, 6.21648645733122950963363322555, 7.17021858082527622158894999152, 8.070664119349607603211823988416, 9.239248097409725813780480092263, 10.44283563240889743046930479483, 10.94112423295165005585658954793