| L(s) = 1 | + 1.65·2-s + 0.722·4-s − 2.19·5-s − 0.319·7-s − 2.10·8-s − 3.62·10-s + 11-s − 1.63·13-s − 0.527·14-s − 4.92·16-s − 3.25·17-s − 7.30·19-s − 1.58·20-s + 1.65·22-s − 0.797·23-s − 0.184·25-s − 2.69·26-s − 0.231·28-s + 7.77·29-s + 8.65·31-s − 3.90·32-s − 5.36·34-s + 0.701·35-s + 8.96·37-s − 12.0·38-s + 4.62·40-s − 12.3·41-s + ⋯ |
| L(s) = 1 | + 1.16·2-s + 0.361·4-s − 0.981·5-s − 0.120·7-s − 0.745·8-s − 1.14·10-s + 0.301·11-s − 0.452·13-s − 0.140·14-s − 1.23·16-s − 0.789·17-s − 1.67·19-s − 0.354·20-s + 0.351·22-s − 0.166·23-s − 0.0369·25-s − 0.527·26-s − 0.0436·28-s + 1.44·29-s + 1.55·31-s − 0.690·32-s − 0.920·34-s + 0.118·35-s + 1.47·37-s − 1.95·38-s + 0.731·40-s − 1.92·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.837041684\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.837041684\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 - T \) |
| 61 | \( 1 + T \) |
| good | 2 | \( 1 - 1.65T + 2T^{2} \) |
| 5 | \( 1 + 2.19T + 5T^{2} \) |
| 7 | \( 1 + 0.319T + 7T^{2} \) |
| 13 | \( 1 + 1.63T + 13T^{2} \) |
| 17 | \( 1 + 3.25T + 17T^{2} \) |
| 19 | \( 1 + 7.30T + 19T^{2} \) |
| 23 | \( 1 + 0.797T + 23T^{2} \) |
| 29 | \( 1 - 7.77T + 29T^{2} \) |
| 31 | \( 1 - 8.65T + 31T^{2} \) |
| 37 | \( 1 - 8.96T + 37T^{2} \) |
| 41 | \( 1 + 12.3T + 41T^{2} \) |
| 43 | \( 1 - 12.1T + 43T^{2} \) |
| 47 | \( 1 - 12.9T + 47T^{2} \) |
| 53 | \( 1 - 5.25T + 53T^{2} \) |
| 59 | \( 1 - 7.12T + 59T^{2} \) |
| 67 | \( 1 + 12.2T + 67T^{2} \) |
| 71 | \( 1 + 4.00T + 71T^{2} \) |
| 73 | \( 1 + 6.19T + 73T^{2} \) |
| 79 | \( 1 - 7.48T + 79T^{2} \) |
| 83 | \( 1 - 12.3T + 83T^{2} \) |
| 89 | \( 1 - 9.74T + 89T^{2} \) |
| 97 | \( 1 + 14.3T + 97T^{2} \) |
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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
−8.125171369223576230379561704892, −7.16866363443039066655315233889, −6.43734805160649379852907188487, −6.01742772533378163354635638133, −4.87960495656768950583411275348, −4.30624873355917379115436331148, −4.02529837848052857176291653099, −2.93178525729200585405209877069, −2.29625663673129787965370417394, −0.56866528885975337217604543694,
0.56866528885975337217604543694, 2.29625663673129787965370417394, 2.93178525729200585405209877069, 4.02529837848052857176291653099, 4.30624873355917379115436331148, 4.87960495656768950583411275348, 6.01742772533378163354635638133, 6.43734805160649379852907188487, 7.16866363443039066655315233889, 8.125171369223576230379561704892
