L(s) = 1 | − 0.724i·3-s + 2.33i·7-s + 2.47·9-s − 2.62·11-s − 4.29i·13-s − 6.85i·17-s − 2.87·19-s + 1.69·21-s + i·23-s − 3.96i·27-s + 5.03·29-s − 7.31·31-s + 1.90i·33-s + 9.24i·37-s − 3.11·39-s + ⋯ |
L(s) = 1 | − 0.418i·3-s + 0.883i·7-s + 0.824·9-s − 0.792·11-s − 1.19i·13-s − 1.66i·17-s − 0.659·19-s + 0.369·21-s + 0.208i·23-s − 0.763i·27-s + 0.935·29-s − 1.31·31-s + 0.331i·33-s + 1.51i·37-s − 0.498·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7861675252\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7861675252\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 23 | \( 1 - iT \) |
good | 3 | \( 1 + 0.724iT - 3T^{2} \) |
| 7 | \( 1 - 2.33iT - 7T^{2} \) |
| 11 | \( 1 + 2.62T + 11T^{2} \) |
| 13 | \( 1 + 4.29iT - 13T^{2} \) |
| 17 | \( 1 + 6.85iT - 17T^{2} \) |
| 19 | \( 1 + 2.87T + 19T^{2} \) |
| 29 | \( 1 - 5.03T + 29T^{2} \) |
| 31 | \( 1 + 7.31T + 31T^{2} \) |
| 37 | \( 1 - 9.24iT - 37T^{2} \) |
| 41 | \( 1 + 6.95T + 41T^{2} \) |
| 43 | \( 1 - 7.01iT - 43T^{2} \) |
| 47 | \( 1 + 4.74iT - 47T^{2} \) |
| 53 | \( 1 + 12.3iT - 53T^{2} \) |
| 59 | \( 1 - 2.14T + 59T^{2} \) |
| 61 | \( 1 + 2.30T + 61T^{2} \) |
| 67 | \( 1 - 1.98iT - 67T^{2} \) |
| 71 | \( 1 + 6.87T + 71T^{2} \) |
| 73 | \( 1 - 13.2iT - 73T^{2} \) |
| 79 | \( 1 + 16.6T + 79T^{2} \) |
| 83 | \( 1 - 9.09iT - 83T^{2} \) |
| 89 | \( 1 + 0.676T + 89T^{2} \) |
| 97 | \( 1 + 15.0iT - 97T^{2} \) |
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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
−8.184608711534520757624610871620, −7.16130639554780205002507434500, −6.78285341533769777834560158662, −5.64883710720481608155154968285, −5.21391532929184479035799970497, −4.39378948114668665695359884183, −3.10461940884640374115077382064, −2.59278825920438505816928972653, −1.50202874673285258162631824811, −0.20992876490978242350656714206,
1.39448779421601952574778022814, 2.21692322938103891377284903477, 3.60320300653802032491917749404, 4.13194174965719809284695288022, 4.68280456828322761680315553675, 5.71964629216174305773415535120, 6.53008552104614100561351817942, 7.22424954975826214322663317443, 7.78807991341935235032045138242, 8.750874479056850884258132921795