| L(s) = 1 | − 2.63·2-s + 4.94·4-s − 5-s − 3.28·7-s − 7.77·8-s + 2.63·10-s + 3.22·11-s + 8.65·14-s + 10.5·16-s + 4.25·17-s − 2.87·19-s − 4.94·20-s − 8.51·22-s − 6.09·23-s + 25-s − 16.2·28-s − 5.77·29-s − 0.835·31-s − 12.3·32-s − 11.2·34-s + 3.28·35-s − 5.59·37-s + 7.57·38-s + 7.77·40-s − 2.18·41-s + 2.48·43-s + 15.9·44-s + ⋯ |
| L(s) = 1 | − 1.86·2-s + 2.47·4-s − 0.447·5-s − 1.24·7-s − 2.74·8-s + 0.833·10-s + 0.973·11-s + 2.31·14-s + 2.64·16-s + 1.03·17-s − 0.659·19-s − 1.10·20-s − 1.81·22-s − 1.27·23-s + 0.200·25-s − 3.06·28-s − 1.07·29-s − 0.150·31-s − 2.18·32-s − 1.92·34-s + 0.554·35-s − 0.919·37-s + 1.22·38-s + 1.22·40-s − 0.341·41-s + 0.379·43-s + 2.40·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3712793378\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3712793378\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 2.63T + 2T^{2} \) |
| 7 | \( 1 + 3.28T + 7T^{2} \) |
| 11 | \( 1 - 3.22T + 11T^{2} \) |
| 17 | \( 1 - 4.25T + 17T^{2} \) |
| 19 | \( 1 + 2.87T + 19T^{2} \) |
| 23 | \( 1 + 6.09T + 23T^{2} \) |
| 29 | \( 1 + 5.77T + 29T^{2} \) |
| 31 | \( 1 + 0.835T + 31T^{2} \) |
| 37 | \( 1 + 5.59T + 37T^{2} \) |
| 41 | \( 1 + 2.18T + 41T^{2} \) |
| 43 | \( 1 - 2.48T + 43T^{2} \) |
| 47 | \( 1 + 10.5T + 47T^{2} \) |
| 53 | \( 1 - 5.08T + 53T^{2} \) |
| 59 | \( 1 - 0.144T + 59T^{2} \) |
| 61 | \( 1 - 6.06T + 61T^{2} \) |
| 67 | \( 1 - 12.6T + 67T^{2} \) |
| 71 | \( 1 - 9.02T + 71T^{2} \) |
| 73 | \( 1 + 5.70T + 73T^{2} \) |
| 79 | \( 1 - 14.1T + 79T^{2} \) |
| 83 | \( 1 + 7.41T + 83T^{2} \) |
| 89 | \( 1 - 13.0T + 89T^{2} \) |
| 97 | \( 1 - 2.97T + 97T^{2} \) |
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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
−8.077475503442057758050652589863, −7.31211079060173452782642227434, −6.66745916786373863358628138207, −6.28898852816244024986330707065, −5.40501181741972304875478132666, −3.85409638703316699191657819511, −3.44834661193111217901422773503, −2.36991616943555532529499696152, −1.49995429870570694499964735308, −0.41128473847620119265947942113,
0.41128473847620119265947942113, 1.49995429870570694499964735308, 2.36991616943555532529499696152, 3.44834661193111217901422773503, 3.85409638703316699191657819511, 5.40501181741972304875478132666, 6.28898852816244024986330707065, 6.66745916786373863358628138207, 7.31211079060173452782642227434, 8.077475503442057758050652589863
