L(s) = 1 | + i·3-s − i·5-s + 0.279·7-s − 9-s + 0.628·11-s − 3.03·13-s + 15-s − 2.36i·17-s − 1.48·19-s + 0.279i·21-s + (−1.08 − 4.67i)23-s − 25-s − i·27-s + 3.72·29-s + 8.99i·31-s + ⋯ |
L(s) = 1 | + 0.577i·3-s − 0.447i·5-s + 0.105·7-s − 0.333·9-s + 0.189·11-s − 0.840·13-s + 0.258·15-s − 0.573i·17-s − 0.341·19-s + 0.0609i·21-s + (−0.226 − 0.974i)23-s − 0.200·25-s − 0.192i·27-s + 0.691·29-s + 1.61i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.226 - 0.974i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.226 - 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.182995506\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.182995506\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 + iT \) |
| 23 | \( 1 + (1.08 + 4.67i)T \) |
good | 7 | \( 1 - 0.279T + 7T^{2} \) |
| 11 | \( 1 - 0.628T + 11T^{2} \) |
| 13 | \( 1 + 3.03T + 13T^{2} \) |
| 17 | \( 1 + 2.36iT - 17T^{2} \) |
| 19 | \( 1 + 1.48T + 19T^{2} \) |
| 29 | \( 1 - 3.72T + 29T^{2} \) |
| 31 | \( 1 - 8.99iT - 31T^{2} \) |
| 37 | \( 1 - 5.99iT - 37T^{2} \) |
| 41 | \( 1 + 7.39T + 41T^{2} \) |
| 43 | \( 1 - 8.69T + 43T^{2} \) |
| 47 | \( 1 - 2.41iT - 47T^{2} \) |
| 53 | \( 1 + 1.22iT - 53T^{2} \) |
| 59 | \( 1 - 7.12iT - 59T^{2} \) |
| 61 | \( 1 - 3.34iT - 61T^{2} \) |
| 67 | \( 1 + 8.37T + 67T^{2} \) |
| 71 | \( 1 + 3.78iT - 71T^{2} \) |
| 73 | \( 1 - 15.1T + 73T^{2} \) |
| 79 | \( 1 - 16.5T + 79T^{2} \) |
| 83 | \( 1 - 7.07T + 83T^{2} \) |
| 89 | \( 1 - 12.8iT - 89T^{2} \) |
| 97 | \( 1 - 12.7iT - 97T^{2} \) |
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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
−8.389253956975886961742437054840, −7.81067114589855464706255941106, −6.81272581384217756393600563764, −6.30293963490573583569682244945, −5.07468951683662584162381917500, −4.92001895677526138486468600797, −4.03820085006013574815825021261, −3.08187285983055178170204415302, −2.27072993205480713990149929763, −1.00527118003519328107305890906,
0.34029791827950984980731671926, 1.73436854261018731109978094622, 2.40278291185133705788836485522, 3.39610162143559747776651149202, 4.19823977985987989328623097761, 5.14242938122155297247203683648, 5.94359448382427117468718823878, 6.53360011229036435782522311669, 7.31216019013528945643421631377, 7.82940845492149002780051065598