L(s) = 1 | + (0.841 + 0.540i)3-s + (0.654 − 0.755i)5-s + (0.654 − 0.755i)7-s + (0.415 + 0.909i)9-s + (−0.654 − 0.755i)11-s + (−0.841 − 0.540i)13-s + (0.959 − 0.281i)15-s + (−0.959 − 0.281i)17-s + (0.415 + 0.909i)19-s + (0.959 − 0.281i)21-s + (−0.415 − 0.909i)23-s + (−0.142 − 0.989i)25-s + (−0.142 + 0.989i)27-s + (0.654 − 0.755i)29-s + (−0.415 + 0.909i)31-s + ⋯ |
L(s) = 1 | + (0.841 + 0.540i)3-s + (0.654 − 0.755i)5-s + (0.654 − 0.755i)7-s + (0.415 + 0.909i)9-s + (−0.654 − 0.755i)11-s + (−0.841 − 0.540i)13-s + (0.959 − 0.281i)15-s + (−0.959 − 0.281i)17-s + (0.415 + 0.909i)19-s + (0.959 − 0.281i)21-s + (−0.415 − 0.909i)23-s + (−0.142 − 0.989i)25-s + (−0.142 + 0.989i)27-s + (0.654 − 0.755i)29-s + (−0.415 + 0.909i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.453 - 0.891i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.453 - 0.891i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.145667751 - 1.867447787i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.145667751 - 1.867447787i\) |
\(L(1)\) |
\(\approx\) |
\(1.374790308 - 0.3074359109i\) |
\(L(1)\) |
\(\approx\) |
\(1.374790308 - 0.3074359109i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 89 | \( 1 \) |
good | 3 | \( 1 + (0.841 + 0.540i)T \) |
| 5 | \( 1 + (0.654 - 0.755i)T \) |
| 7 | \( 1 + (0.654 - 0.755i)T \) |
| 11 | \( 1 + (-0.654 - 0.755i)T \) |
| 13 | \( 1 + (-0.841 - 0.540i)T \) |
| 17 | \( 1 + (-0.959 - 0.281i)T \) |
| 19 | \( 1 + (0.415 + 0.909i)T \) |
| 23 | \( 1 + (-0.415 - 0.909i)T \) |
| 29 | \( 1 + (0.654 - 0.755i)T \) |
| 31 | \( 1 + (-0.415 + 0.909i)T \) |
| 37 | \( 1 - T \) |
| 41 | \( 1 + (0.841 - 0.540i)T \) |
| 43 | \( 1 + (-0.654 - 0.755i)T \) |
| 47 | \( 1 + (-0.841 + 0.540i)T \) |
| 53 | \( 1 + (-0.841 - 0.540i)T \) |
| 59 | \( 1 + (0.841 - 0.540i)T \) |
| 61 | \( 1 + (0.142 - 0.989i)T \) |
| 67 | \( 1 + (0.841 + 0.540i)T \) |
| 71 | \( 1 + (0.654 + 0.755i)T \) |
| 73 | \( 1 + (0.415 - 0.909i)T \) |
| 79 | \( 1 + (-0.415 + 0.909i)T \) |
| 83 | \( 1 + (-0.959 - 0.281i)T \) |
| 97 | \( 1 + (-0.654 + 0.755i)T \) |
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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
−22.5165189035951607908899896863, −21.64468482005812025814237059192, −21.207694441612219459586760271011, −20.09429207921443461019477641267, −19.44549875665356317798351203771, −18.45228286907103472693702073600, −17.9079288122264557734691730605, −17.41736757778758114324149780413, −15.74832767107762690725944644241, −15.05271827377774249143006509547, −14.4935800906856940959059477016, −13.62401992482971907611128814929, −12.89010830757732174518626467063, −11.9076024735460710350685598793, −11.02770030338672103504174741080, −9.79620113383176039564803793326, −9.27123244746949200234610348637, −8.22170272356324301108559241080, −7.29000381941526925207980904988, −6.67270890126781523974947730924, −5.46771692601402413025572696418, −4.46679585440676572916525328390, −3.00112935182226771133189433463, −2.27410244963466346454715916076, −1.66020355333391670987682711083,
0.38385266123273925259099852092, 1.73926739823677414195474754546, 2.651805122149036588813445432338, 3.85321072084740102071077534051, 4.82860964299782336802690053933, 5.400717627429082790951282330314, 6.8674971321032879890892685779, 8.13266892049452681165685391145, 8.361411866045111743010642637722, 9.58184119105845531696415373888, 10.25291467526659499755991181883, 10.98024118452296058945393256443, 12.35981057978721381783521657132, 13.222741371434469484345146857465, 14.017051843450883726797834477023, 14.42820730598109620066660269014, 15.721383411365024642108693351753, 16.266828570987037171360291085540, 17.21472611745227633979579736046, 17.92856191271931702016841909006, 19.03481057952180673044522260675, 20.04498953028746304930963770378, 20.49885128044525076601389994376, 21.15983166538622061248711010354, 21.88079948363961385631877262011