L(s) = 1 | + (−1.67 − 1.67i)2-s + (−2.55 − 1.56i)3-s + 1.58i·4-s + (0.529 − 4.97i)5-s + (1.66 + 6.89i)6-s + (−1.73 − 6.78i)7-s + (−4.03 + 4.03i)8-s + (4.10 + 8.01i)9-s + (−9.19 + 7.42i)10-s + 4.41i·11-s + (2.48 − 4.06i)12-s + (−1.62 + 1.62i)13-s + (−8.43 + 14.2i)14-s + (−9.13 + 11.8i)15-s + 19.8·16-s + (−13.9 + 13.9i)17-s + ⋯ |
L(s) = 1 | + (−0.835 − 0.835i)2-s + (−0.853 − 0.521i)3-s + 0.397i·4-s + (0.105 − 0.994i)5-s + (0.277 + 1.14i)6-s + (−0.247 − 0.968i)7-s + (−0.503 + 0.503i)8-s + (0.455 + 0.890i)9-s + (−0.919 + 0.742i)10-s + 0.401i·11-s + (0.207 − 0.338i)12-s + (−0.124 + 0.124i)13-s + (−0.602 + 1.01i)14-s + (−0.609 + 0.793i)15-s + 1.23·16-s + (−0.819 + 0.819i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.575 - 0.817i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.575 - 0.817i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.160985 + 0.310044i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.160985 + 0.310044i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (2.55 + 1.56i)T \) |
| 5 | \( 1 + (-0.529 + 4.97i)T \) |
| 7 | \( 1 + (1.73 + 6.78i)T \) |
good | 2 | \( 1 + (1.67 + 1.67i)T + 4iT^{2} \) |
| 11 | \( 1 - 4.41iT - 121T^{2} \) |
| 13 | \( 1 + (1.62 - 1.62i)T - 169iT^{2} \) |
| 17 | \( 1 + (13.9 - 13.9i)T - 289iT^{2} \) |
| 19 | \( 1 - 0.694T + 361T^{2} \) |
| 23 | \( 1 + (-23.1 + 23.1i)T - 529iT^{2} \) |
| 29 | \( 1 + 49.1T + 841T^{2} \) |
| 31 | \( 1 + 33.8iT - 961T^{2} \) |
| 37 | \( 1 + (-2.02 + 2.02i)T - 1.36e3iT^{2} \) |
| 41 | \( 1 - 32.5T + 1.68e3T^{2} \) |
| 43 | \( 1 + (30.4 + 30.4i)T + 1.84e3iT^{2} \) |
| 47 | \( 1 + (18.7 - 18.7i)T - 2.20e3iT^{2} \) |
| 53 | \( 1 + (-33.8 + 33.8i)T - 2.80e3iT^{2} \) |
| 59 | \( 1 - 23.1iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 12.9iT - 3.72e3T^{2} \) |
| 67 | \( 1 + (56.3 - 56.3i)T - 4.48e3iT^{2} \) |
| 71 | \( 1 + 92.7iT - 5.04e3T^{2} \) |
| 73 | \( 1 + (95.4 - 95.4i)T - 5.32e3iT^{2} \) |
| 79 | \( 1 + 100. iT - 6.24e3T^{2} \) |
| 83 | \( 1 + (-5.62 - 5.62i)T + 6.88e3iT^{2} \) |
| 89 | \( 1 + 158. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (-37.1 - 37.1i)T + 9.40e3iT^{2} \) |
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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
−12.74175306027806147065590553577, −11.57977056706389056052912770115, −10.74658218858136199977862832114, −9.819735967698617781136490672752, −8.677641311395851419730042354154, −7.36128328012560576464405924877, −5.90768813814127468623041594032, −4.44975740200054676809720640363, −1.79623594621382108552937919350, −0.37285198986103562626621932439,
3.26415836957764839987433581959, 5.44301377900046536254577196426, 6.45280194335087387941614805239, 7.36255248785204599052831342027, 8.990776600391247484988418422238, 9.668577674534834344189780786135, 10.96504744366471607755964925742, 11.80539364595912122148649003734, 13.10163553925817250628821936237, 14.79419747412341624435027622687