L(s) = 1 | + (−0.629 + 1.26i)2-s + (−1.20 − 1.59i)4-s + 2.32i·5-s + (2.77 − 0.524i)8-s + (−2.94 − 1.46i)10-s + 3.58i·11-s − 2.93i·13-s + (−1.08 + 3.84i)16-s + 2.32i·17-s + 8.33·19-s + (3.71 − 2.80i)20-s + (−4.53 − 2.25i)22-s + 1.48i·23-s − 0.414·25-s + (3.71 + 1.84i)26-s + ⋯ |
L(s) = 1 | + (−0.445 + 0.895i)2-s + (−0.603 − 0.797i)4-s + 1.04i·5-s + (0.982 − 0.185i)8-s + (−0.931 − 0.463i)10-s + 1.07i·11-s − 0.812i·13-s + (−0.271 + 0.962i)16-s + 0.564i·17-s + 1.91·19-s + (0.829 − 0.628i)20-s + (−0.966 − 0.480i)22-s + 0.309i·23-s − 0.0828·25-s + (0.727 + 0.361i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.876 - 0.480i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.876 - 0.480i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.178142335\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.178142335\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.629 - 1.26i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 - 2.32iT - 5T^{2} \) |
| 11 | \( 1 - 3.58iT - 11T^{2} \) |
| 13 | \( 1 + 2.93iT - 13T^{2} \) |
| 17 | \( 1 - 2.32iT - 17T^{2} \) |
| 19 | \( 1 - 8.33T + 19T^{2} \) |
| 23 | \( 1 - 1.48iT - 23T^{2} \) |
| 29 | \( 1 - 7.86T + 29T^{2} \) |
| 31 | \( 1 + 3.45T + 31T^{2} \) |
| 37 | \( 1 + 8.24T + 37T^{2} \) |
| 41 | \( 1 + 2.89iT - 41T^{2} \) |
| 43 | \( 1 - 6.37iT - 43T^{2} \) |
| 47 | \( 1 + 10.4T + 47T^{2} \) |
| 53 | \( 1 - 8.59T + 53T^{2} \) |
| 59 | \( 1 - 10.4T + 59T^{2} \) |
| 61 | \( 1 - 2.93iT - 61T^{2} \) |
| 67 | \( 1 - 9.02iT - 67T^{2} \) |
| 71 | \( 1 - 10.7iT - 71T^{2} \) |
| 73 | \( 1 + 11.2iT - 73T^{2} \) |
| 79 | \( 1 - 15.3iT - 79T^{2} \) |
| 83 | \( 1 + 14.8T + 83T^{2} \) |
| 89 | \( 1 - 13.5iT - 89T^{2} \) |
| 97 | \( 1 + 5.35iT - 97T^{2} \) |
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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
−9.872291913073605848165500975714, −8.684405246809536105005836572886, −7.912667278765075190979013198708, −7.10989710934573631268082319089, −6.79485173061306347931247497378, −5.64470734464818337976648445888, −5.04564594488092091591210661088, −3.82785206165265108736648119666, −2.75826620203352041647439127719, −1.30806444956721212805661182589,
0.59517265524206225672799480948, 1.53595776900806944372834322203, 2.90571027985495000956961917727, 3.72245473939323832177918032989, 4.84139191975332147598085315900, 5.34877771586099537551878771339, 6.73038537674487359279044752966, 7.64578400356047961743175022877, 8.592839370736878554402694791104, 8.891093246936704412727241888950