L(s) = 1 | − 1.88i·2-s − 2.43·3-s − 1.53·4-s + 2.91·5-s + 4.58i·6-s − 0.518i·7-s − 0.875i·8-s + 2.94·9-s − 5.48i·10-s − 2.72i·11-s + 3.74·12-s − 2.46i·13-s − 0.974·14-s − 7.11·15-s − 4.71·16-s + 2.36·17-s + ⋯ |
L(s) = 1 | − 1.32i·2-s − 1.40·3-s − 0.767·4-s + 1.30·5-s + 1.87i·6-s − 0.195i·7-s − 0.309i·8-s + 0.982·9-s − 1.73i·10-s − 0.821i·11-s + 1.08·12-s − 0.684i·13-s − 0.260·14-s − 1.83·15-s − 1.17·16-s + 0.574·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 349 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.969 + 0.245i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 349 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.969 + 0.245i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.117053 - 0.937369i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.117053 - 0.937369i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 349 | \( 1 + (-18.1 + 4.59i)T \) |
good | 2 | \( 1 + 1.88iT - 2T^{2} \) |
| 3 | \( 1 + 2.43T + 3T^{2} \) |
| 5 | \( 1 - 2.91T + 5T^{2} \) |
| 7 | \( 1 + 0.518iT - 7T^{2} \) |
| 11 | \( 1 + 2.72iT - 11T^{2} \) |
| 13 | \( 1 + 2.46iT - 13T^{2} \) |
| 17 | \( 1 - 2.36T + 17T^{2} \) |
| 19 | \( 1 + 2.12T + 19T^{2} \) |
| 23 | \( 1 + 4.45T + 23T^{2} \) |
| 29 | \( 1 - 2.69T + 29T^{2} \) |
| 31 | \( 1 + 2.08T + 31T^{2} \) |
| 37 | \( 1 + 5.09T + 37T^{2} \) |
| 41 | \( 1 + 0.0671T + 41T^{2} \) |
| 43 | \( 1 + 10.2iT - 43T^{2} \) |
| 47 | \( 1 + 7.54iT - 47T^{2} \) |
| 53 | \( 1 + 3.87iT - 53T^{2} \) |
| 59 | \( 1 - 12.6iT - 59T^{2} \) |
| 61 | \( 1 + 0.494iT - 61T^{2} \) |
| 67 | \( 1 - 12.7T + 67T^{2} \) |
| 71 | \( 1 - 7.68iT - 71T^{2} \) |
| 73 | \( 1 - 9.52T + 73T^{2} \) |
| 79 | \( 1 - 4.86iT - 79T^{2} \) |
| 83 | \( 1 - 14.6T + 83T^{2} \) |
| 89 | \( 1 - 13.4iT - 89T^{2} \) |
| 97 | \( 1 + 7.88iT - 97T^{2} \) |
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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
−10.94431495276121140150944893739, −10.38456777100327059142696114322, −9.884858254209472563672065251715, −8.620457907348050311919557441201, −6.87537098564186851889230914341, −5.91390404359811005380321793664, −5.25043716816878201541943890809, −3.70535823199977090775736217959, −2.20583931060694705205158782613, −0.78090399962347144177155440748,
1.98679864968530948636915765941, 4.63335172612044581532325928108, 5.41696485162132936473721098485, 6.22311275753460330679530330913, 6.65534203514763439953209803904, 7.84492248107254180916676266134, 9.165786117014134743435314690720, 9.999529728436735690196442287476, 10.95290530513276974752891293037, 11.96513094384344082917629184242