| L(s) = 1 | + (−1.31 − 1.12i)3-s + 5-s + (2.64 + 0.0644i)7-s + (0.468 + 2.96i)9-s + 2.69i·11-s + 6.61i·13-s + (−1.31 − 1.12i)15-s − 7.03·17-s − 3.22i·19-s + (−3.41 − 3.06i)21-s + 2.56i·23-s + 25-s + (2.71 − 4.42i)27-s + 8.21i·29-s + 2.87i·31-s + ⋯ |
| L(s) = 1 | + (−0.760 − 0.649i)3-s + 0.447·5-s + (0.999 + 0.0243i)7-s + (0.156 + 0.987i)9-s + 0.813i·11-s + 1.83i·13-s + (−0.340 − 0.290i)15-s − 1.70·17-s − 0.740i·19-s + (−0.744 − 0.667i)21-s + 0.534i·23-s + 0.200·25-s + (0.522 − 0.852i)27-s + 1.52i·29-s + 0.515i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.667 - 0.744i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.667 - 0.744i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.12462 + 0.501834i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.12462 + 0.501834i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.31 + 1.12i)T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + (-2.64 - 0.0644i)T \) |
| good | 11 | \( 1 - 2.69iT - 11T^{2} \) |
| 13 | \( 1 - 6.61iT - 13T^{2} \) |
| 17 | \( 1 + 7.03T + 17T^{2} \) |
| 19 | \( 1 + 3.22iT - 19T^{2} \) |
| 23 | \( 1 - 2.56iT - 23T^{2} \) |
| 29 | \( 1 - 8.21iT - 29T^{2} \) |
| 31 | \( 1 - 2.87iT - 31T^{2} \) |
| 37 | \( 1 + 6.79T + 37T^{2} \) |
| 41 | \( 1 - 10.0T + 41T^{2} \) |
| 43 | \( 1 - 11.6T + 43T^{2} \) |
| 47 | \( 1 + 2.56T + 47T^{2} \) |
| 53 | \( 1 - 2.70iT - 53T^{2} \) |
| 59 | \( 1 + 3.03T + 59T^{2} \) |
| 61 | \( 1 - 7.83iT - 61T^{2} \) |
| 67 | \( 1 + 12.5T + 67T^{2} \) |
| 71 | \( 1 + 14.0iT - 71T^{2} \) |
| 73 | \( 1 + 1.81iT - 73T^{2} \) |
| 79 | \( 1 - 13.4T + 79T^{2} \) |
| 83 | \( 1 - 3.01T + 83T^{2} \) |
| 89 | \( 1 - 4.17T + 89T^{2} \) |
| 97 | \( 1 + 9.55iT - 97T^{2} \) |
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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
−10.73938214423822345110008348218, −9.244023877711967651933236686062, −8.822823598051475518741843555197, −7.37975499810926030291337467011, −6.97575721551740751867316414247, −6.05808672856456126481685863938, −4.84120531141056200031652195794, −4.44891857237084435655803671782, −2.24907776694845667041480579518, −1.57024148974815046045400487283,
0.68457143292057628547454594833, 2.45618729822663606420324956610, 3.87725381992913806557462121079, 4.80884876471090509127568909695, 5.72255782324912666787784129267, 6.23315346730960160511396275034, 7.63805133647593263902978602217, 8.453203422099959346770538163379, 9.312383143695041044526101906492, 10.38076514864062188684063605338
