L(s) = 1 | + (−0.245 − 0.969i)2-s + (−0.879 + 0.475i)4-s + (−0.879 + 0.475i)5-s + (0.677 + 0.735i)8-s + (0.677 + 0.735i)10-s + (−0.401 + 0.915i)11-s + (0.0825 − 0.996i)13-s + (0.546 − 0.837i)16-s + (0.789 + 0.614i)17-s + (−0.677 + 0.735i)19-s + (0.546 − 0.837i)20-s + (0.986 + 0.164i)22-s + (0.677 − 0.735i)23-s + (0.546 − 0.837i)25-s + (−0.986 + 0.164i)26-s + ⋯ |
L(s) = 1 | + (−0.245 − 0.969i)2-s + (−0.879 + 0.475i)4-s + (−0.879 + 0.475i)5-s + (0.677 + 0.735i)8-s + (0.677 + 0.735i)10-s + (−0.401 + 0.915i)11-s + (0.0825 − 0.996i)13-s + (0.546 − 0.837i)16-s + (0.789 + 0.614i)17-s + (−0.677 + 0.735i)19-s + (0.546 − 0.837i)20-s + (0.986 + 0.164i)22-s + (0.677 − 0.735i)23-s + (0.546 − 0.837i)25-s + (−0.986 + 0.164i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4011 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.368 - 0.929i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4011 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.368 - 0.929i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9181255417 - 0.6237675637i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9181255417 - 0.6237675637i\) |
\(L(1)\) |
\(\approx\) |
\(0.6991806758 - 0.2108528420i\) |
\(L(1)\) |
\(\approx\) |
\(0.6991806758 - 0.2108528420i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| 191 | \( 1 \) |
good | 2 | \( 1 + (-0.245 - 0.969i)T \) |
| 5 | \( 1 + (-0.879 + 0.475i)T \) |
| 11 | \( 1 + (-0.401 + 0.915i)T \) |
| 13 | \( 1 + (0.0825 - 0.996i)T \) |
| 17 | \( 1 + (0.789 + 0.614i)T \) |
| 19 | \( 1 + (-0.677 + 0.735i)T \) |
| 23 | \( 1 + (0.677 - 0.735i)T \) |
| 29 | \( 1 + (0.546 + 0.837i)T \) |
| 31 | \( 1 + (-0.986 + 0.164i)T \) |
| 37 | \( 1 + (0.986 + 0.164i)T \) |
| 41 | \( 1 + (-0.245 + 0.969i)T \) |
| 43 | \( 1 + (0.945 - 0.324i)T \) |
| 47 | \( 1 + (0.401 - 0.915i)T \) |
| 53 | \( 1 + (-0.401 + 0.915i)T \) |
| 59 | \( 1 + (-0.986 + 0.164i)T \) |
| 61 | \( 1 + (0.789 - 0.614i)T \) |
| 67 | \( 1 + (0.789 - 0.614i)T \) |
| 71 | \( 1 + (0.245 - 0.969i)T \) |
| 73 | \( 1 + (-0.401 - 0.915i)T \) |
| 79 | \( 1 + (0.546 - 0.837i)T \) |
| 83 | \( 1 + (0.677 + 0.735i)T \) |
| 89 | \( 1 + (-0.945 - 0.324i)T \) |
| 97 | \( 1 + (0.986 + 0.164i)T \) |
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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
−18.56749853670264856486632150362, −17.53171817345572888753746701435, −16.96891390684690276041819601420, −16.21706878525911111283688271791, −15.95138240579743815774507488659, −15.200717486632671844618449774307, −14.44443740002104717558520565762, −13.8020641180786407380821352893, −13.09082722220817226617637199260, −12.433345574785965750088139471782, −11.333448368660373549245220956508, −11.07072358170387537440810434263, −9.86911010341392326278796008358, −9.1492277050541036166554067042, −8.66980815129245182098590022526, −7.877053677526491440210284792243, −7.34828962801005641819000678092, −6.60131499396341264449637068129, −5.69753076362263238399986775518, −5.082732632020965606220871605513, −4.264069210275136402098257946782, −3.6788181253339809097941625669, −2.57911083214303680867871885063, −1.182541065623476122056700293153, −0.545317789822358811803576517129,
0.35484764917104918409477378976, 1.28493540634199496647626440351, 2.27301358053919531420182272602, 3.062540887951566525846281823815, 3.64540510550057108311193112873, 4.46796802299973148849207910597, 5.13796678890587976664267580000, 6.18278635833083832177348558345, 7.24652388208590938571325513131, 7.87923539271065950210783627389, 8.340555923051561508424672088527, 9.26095642198868623191186477360, 10.18934226992731620784262943165, 10.61359059492424149163208023531, 11.08328579581264837248463228987, 12.23956322933777066761706861190, 12.44903374540523367418238224708, 13.023373084506990033057037633914, 14.12936291222001638950161229484, 14.84744337169541273307715814770, 15.20215921447114821746504910076, 16.3278256006609399092029539958, 16.88549930890373847371902030402, 17.764223258263459056378072283527, 18.39451319209635934367476879003