L(s) = 1 | + i·2-s + i·3-s − 4-s − 6-s − i·8-s − 9-s + 11-s − i·12-s + 4i·13-s + 16-s − 4i·17-s − i·18-s + 19-s + i·22-s + 5i·23-s + 24-s + ⋯ |
L(s) = 1 | + 0.707i·2-s + 0.577i·3-s − 0.5·4-s − 0.408·6-s − 0.353i·8-s − 0.333·9-s + 0.301·11-s − 0.288i·12-s + 1.10i·13-s + 0.250·16-s − 0.970i·17-s − 0.235i·18-s + 0.229·19-s + 0.213i·22-s + 1.04i·23-s + 0.204·24-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8914620275\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8914620275\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - iT \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 \) |
| 19 | \( 1 - T \) |
good | 7 | \( 1 - 7T^{2} \) |
| 11 | \( 1 - T + 11T^{2} \) |
| 13 | \( 1 - 4iT - 13T^{2} \) |
| 17 | \( 1 + 4iT - 17T^{2} \) |
| 23 | \( 1 - 5iT - 23T^{2} \) |
| 29 | \( 1 + 3T + 29T^{2} \) |
| 31 | \( 1 + 5T + 31T^{2} \) |
| 37 | \( 1 - 6iT - 37T^{2} \) |
| 41 | \( 1 + 2T + 41T^{2} \) |
| 43 | \( 1 - 4iT - 43T^{2} \) |
| 47 | \( 1 - 47T^{2} \) |
| 53 | \( 1 - 9iT - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 11T + 61T^{2} \) |
| 67 | \( 1 - iT - 67T^{2} \) |
| 71 | \( 1 - 2T + 71T^{2} \) |
| 73 | \( 1 + 3iT - 73T^{2} \) |
| 79 | \( 1 + 17T + 79T^{2} \) |
| 83 | \( 1 - 3iT - 83T^{2} \) |
| 89 | \( 1 + 7T + 89T^{2} \) |
| 97 | \( 1 + 10iT - 97T^{2} \) |
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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
−9.319685545573095156782251187860, −8.557966385680972589698618709678, −7.54011624280925526415780954553, −7.05063172385998476454132587082, −6.12708863969531851006319969914, −5.38732582572456385147962861786, −4.59315042880377684866519876149, −3.88267159542914329155661524632, −2.90577199714807317140501161795, −1.47926245411887905522198437659,
0.28934424354243803265461894676, 1.49984459642685948920931233809, 2.44201825057909405270727932409, 3.40905153845812868487171191736, 4.18519511194129856097562017301, 5.34694383631505401796429914047, 5.91384900513542665602240656917, 6.90706050980109179698988188690, 7.70251644911617365019374728121, 8.435769313011631165278233580013