L(s) = 1 | − i·2-s + (1.68 − 0.421i)3-s − 4-s − 5-s + (−0.421 − 1.68i)6-s + i·8-s + (2.64 − 1.41i)9-s + i·10-s + 0.193i·11-s + (−1.68 + 0.421i)12-s + 1.54i·13-s + (−1.68 + 0.421i)15-s + 16-s − 0.529·17-s + (−1.41 − 2.64i)18-s − 6.38i·19-s + ⋯ |
L(s) = 1 | − 0.707i·2-s + (0.969 − 0.243i)3-s − 0.5·4-s − 0.447·5-s + (−0.171 − 0.685i)6-s + 0.353i·8-s + (0.881 − 0.471i)9-s + 0.316i·10-s + 0.0584i·11-s + (−0.484 + 0.121i)12-s + 0.429i·13-s + (−0.433 + 0.108i)15-s + 0.250·16-s − 0.128·17-s + (−0.333 − 0.623i)18-s − 1.46i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.451 + 0.892i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.451 + 0.892i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.955343934\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.955343934\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + iT \) |
| 3 | \( 1 + (-1.68 + 0.421i)T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
good | 11 | \( 1 - 0.193iT - 11T^{2} \) |
| 13 | \( 1 - 1.54iT - 13T^{2} \) |
| 17 | \( 1 + 0.529T + 17T^{2} \) |
| 19 | \( 1 + 6.38iT - 19T^{2} \) |
| 23 | \( 1 + 4.24iT - 23T^{2} \) |
| 29 | \( 1 + 4.87iT - 29T^{2} \) |
| 31 | \( 1 + 9.26iT - 31T^{2} \) |
| 37 | \( 1 - 1.76T + 37T^{2} \) |
| 41 | \( 1 - 9.91T + 41T^{2} \) |
| 43 | \( 1 - 11.4T + 43T^{2} \) |
| 47 | \( 1 + 9.80T + 47T^{2} \) |
| 53 | \( 1 + 0.0649iT - 53T^{2} \) |
| 59 | \( 1 + 12.0T + 59T^{2} \) |
| 61 | \( 1 - 4.28iT - 61T^{2} \) |
| 67 | \( 1 - 4.83T + 67T^{2} \) |
| 71 | \( 1 - 6.29iT - 71T^{2} \) |
| 73 | \( 1 - 8.09iT - 73T^{2} \) |
| 79 | \( 1 - 6.77T + 79T^{2} \) |
| 83 | \( 1 + 2.11T + 83T^{2} \) |
| 89 | \( 1 + 16.3T + 89T^{2} \) |
| 97 | \( 1 + 8.24iT - 97T^{2} \) |
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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
−9.336631937034073273789716179499, −8.545976728618441246119780414132, −7.80232906853180751930544129159, −7.04222120184231400218019245308, −6.02761403989003506054357287349, −4.49647922399568231635181756127, −4.14180652348932818434303393783, −2.86075035584117557888130197600, −2.24938023863459792813048741029, −0.73952014630070382292904838763,
1.48072022505784146569812075703, 3.04106827765260995676397310102, 3.76716949141131919502643551897, 4.69988033968594085825069397385, 5.63683179841415567611352428792, 6.68550050784857752029640431404, 7.67242704475198325924217685560, 7.954480847234203763518264114285, 8.892224447851022055525209460303, 9.485118385930719315486836801308