L(s) = 1 | + (−0.342 + 0.939i)5-s + (−0.5 − 0.866i)7-s + i·11-s + (0.342 + 0.939i)13-s + (−0.939 − 0.342i)17-s + (0.173 − 0.984i)23-s + (−0.766 − 0.642i)25-s + (−0.984 − 0.173i)29-s + 31-s + (0.984 − 0.173i)35-s − i·37-s + (0.766 − 0.642i)41-s + (−0.984 + 0.173i)43-s + (−0.173 + 0.984i)47-s + (−0.5 + 0.866i)49-s + ⋯ |
L(s) = 1 | + (−0.342 + 0.939i)5-s + (−0.5 − 0.866i)7-s + i·11-s + (0.342 + 0.939i)13-s + (−0.939 − 0.342i)17-s + (0.173 − 0.984i)23-s + (−0.766 − 0.642i)25-s + (−0.984 − 0.173i)29-s + 31-s + (0.984 − 0.173i)35-s − i·37-s + (0.766 − 0.642i)41-s + (−0.984 + 0.173i)43-s + (−0.173 + 0.984i)47-s + (−0.5 + 0.866i)49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.626 - 0.779i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.626 - 0.779i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.06388218649 - 0.1334142458i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.06388218649 - 0.1334142458i\) |
\(L(1)\) |
\(\approx\) |
\(0.7606937762 + 0.1080633386i\) |
\(L(1)\) |
\(\approx\) |
\(0.7606937762 + 0.1080633386i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 \) |
good | 5 | \( 1 + (-0.342 + 0.939i)T \) |
| 7 | \( 1 + (-0.5 - 0.866i)T \) |
| 11 | \( 1 + iT \) |
| 13 | \( 1 + (0.342 + 0.939i)T \) |
| 17 | \( 1 + (-0.939 - 0.342i)T \) |
| 23 | \( 1 + (0.173 - 0.984i)T \) |
| 29 | \( 1 + (-0.984 - 0.173i)T \) |
| 31 | \( 1 + T \) |
| 37 | \( 1 - iT \) |
| 41 | \( 1 + (0.766 - 0.642i)T \) |
| 43 | \( 1 + (-0.984 + 0.173i)T \) |
| 47 | \( 1 + (-0.173 + 0.984i)T \) |
| 53 | \( 1 + (-0.642 + 0.766i)T \) |
| 59 | \( 1 + (0.984 - 0.173i)T \) |
| 61 | \( 1 + (-0.342 - 0.939i)T \) |
| 67 | \( 1 + (-0.642 + 0.766i)T \) |
| 71 | \( 1 + (-0.766 + 0.642i)T \) |
| 73 | \( 1 + (-0.173 - 0.984i)T \) |
| 79 | \( 1 + (-0.939 - 0.342i)T \) |
| 83 | \( 1 + (-0.866 + 0.5i)T \) |
| 89 | \( 1 + (0.173 - 0.984i)T \) |
| 97 | \( 1 + (-0.766 + 0.642i)T \) |
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\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−19.599644430805474849035924383136, −18.92403696337160910988674723478, −18.07423867543782950099300145791, −17.42316387673391013725680324614, −16.54231155275820653983348122726, −15.96981663777416559527758997386, −15.42776219229561270290405580327, −14.78012978459056793611107639418, −13.44204962588610551934072871991, −13.212867491632628397921994787659, −12.482871165088720316253996432, −11.611800024208655312787645531179, −11.13990094120607392081476215045, −10.077703996050784777043192738990, −9.24675525345045386110383905006, −8.62154139193004698766595422250, −8.168909127693801322651207508972, −7.1725602021710109996884285749, −6.02627647113105076297115837025, −5.658871793556353656816656731, −4.82255646967821713772962166651, −3.76459206773196688634894180526, −3.160881158314580111561065623121, −2.10523568196314123137051845472, −1.04795828444304055069610551898,
0.05008217764107741226424071084, 1.51531314016559602273617268782, 2.474830131612262779233767158642, 3.25037572259370790253371048222, 4.344714629270453379012801039941, 4.48011988559127090750647445432, 6.06698168959709002951678395598, 6.74375063072075118460468395932, 7.12022259444680870718265222598, 7.94648919422519250084580353977, 8.96982266273597085811071167670, 9.78781464620161757728962621824, 10.36140654366925113246187947143, 11.149788098481303656203180667348, 11.69401872512507512763955463626, 12.66519954347471780529634095003, 13.3928891362910189937746005128, 14.09838179659895273886035488100, 14.71521995148974357691435028122, 15.52190656250461440913953454416, 16.08516036928601375165374652557, 16.976224559220771063033137714203, 17.578679180964067484593333809976, 18.4119168764125624616057660878, 18.995602688229568463621811041693