L(s) = 1 | + (−1.36 − 0.366i)2-s + (1.73 + i)4-s − 0.732·7-s + (−1.99 − 2i)8-s + 2i·11-s + 3.46i·13-s + (1 + 0.267i)14-s + (1.99 + 3.46i)16-s + 3.46·17-s + 0.535i·19-s + (0.732 − 2.73i)22-s − 6.19·23-s + (1.26 − 4.73i)26-s + (−1.26 − 0.732i)28-s − 6.92i·29-s + ⋯ |
L(s) = 1 | + (−0.965 − 0.258i)2-s + (0.866 + 0.5i)4-s − 0.276·7-s + (−0.707 − 0.707i)8-s + 0.603i·11-s + 0.960i·13-s + (0.267 + 0.0716i)14-s + (0.499 + 0.866i)16-s + 0.840·17-s + 0.122i·19-s + (0.156 − 0.582i)22-s − 1.29·23-s + (0.248 − 0.928i)26-s + (−0.239 − 0.138i)28-s − 1.28i·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4088336811\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4088336811\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.36 + 0.366i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + 0.732T + 7T^{2} \) |
| 11 | \( 1 - 2iT - 11T^{2} \) |
| 13 | \( 1 - 3.46iT - 13T^{2} \) |
| 17 | \( 1 - 3.46T + 17T^{2} \) |
| 19 | \( 1 - 0.535iT - 19T^{2} \) |
| 23 | \( 1 + 6.19T + 23T^{2} \) |
| 29 | \( 1 + 6.92iT - 29T^{2} \) |
| 31 | \( 1 + 5.46T + 31T^{2} \) |
| 37 | \( 1 - 2iT - 37T^{2} \) |
| 41 | \( 1 + 1.46T + 41T^{2} \) |
| 43 | \( 1 - 5.26iT - 43T^{2} \) |
| 47 | \( 1 - 3.26T + 47T^{2} \) |
| 53 | \( 1 - 11.4iT - 53T^{2} \) |
| 59 | \( 1 - 7.46iT - 59T^{2} \) |
| 61 | \( 1 + 8.92iT - 61T^{2} \) |
| 67 | \( 1 + 10.7iT - 67T^{2} \) |
| 71 | \( 1 + 5.46T + 71T^{2} \) |
| 73 | \( 1 + 7.46T + 73T^{2} \) |
| 79 | \( 1 + 1.07T + 79T^{2} \) |
| 83 | \( 1 - 1.26iT - 83T^{2} \) |
| 89 | \( 1 + 8.92T + 89T^{2} \) |
| 97 | \( 1 - 14.3T + 97T^{2} \) |
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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
−9.623883639691834196312563946893, −8.947625053477352915483551793877, −7.939960560995314752770575416385, −7.47311293747746769888696871426, −6.49950761551401451318023946466, −5.85745124893987466333253314909, −4.45019683532992169697799807571, −3.56501421835816472023405821624, −2.40869432854448879509553710864, −1.47112066662237462857561876752,
0.20843093892489246251001890974, 1.54681971372107638618552544593, 2.84025344071984475779543777462, 3.69654866653822126223870902057, 5.33431407809610082571949379537, 5.77194721436323308792830874430, 6.76402669666804394415958236972, 7.53225238808586563492069854469, 8.258618941157256861134978146173, 8.870598007767415978440369909838