| L(s) = 1 | + 1.39·2-s − 1.65·3-s − 0.0513·4-s − 5-s − 2.31·6-s + 4.70·7-s − 2.86·8-s − 0.259·9-s − 1.39·10-s + 0.0850·12-s + 5.05·13-s + 6.57·14-s + 1.65·15-s − 3.89·16-s + 5.31·17-s − 0.362·18-s + 2.25·19-s + 0.0513·20-s − 7.79·21-s + 1.05·23-s + 4.74·24-s + 25-s + 7.05·26-s + 5.39·27-s − 0.241·28-s + 2.79·29-s + 2.31·30-s + ⋯ |
| L(s) = 1 | + 0.987·2-s − 0.955·3-s − 0.0256·4-s − 0.447·5-s − 0.943·6-s + 1.77·7-s − 1.01·8-s − 0.0865·9-s − 0.441·10-s + 0.0245·12-s + 1.40·13-s + 1.75·14-s + 0.427·15-s − 0.973·16-s + 1.28·17-s − 0.0853·18-s + 0.518·19-s + 0.0114·20-s − 1.70·21-s + 0.219·23-s + 0.967·24-s + 0.200·25-s + 1.38·26-s + 1.03·27-s − 0.0456·28-s + 0.518·29-s + 0.421·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.751740694\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.751740694\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 1.39T + 2T^{2} \) |
| 3 | \( 1 + 1.65T + 3T^{2} \) |
| 7 | \( 1 - 4.70T + 7T^{2} \) |
| 13 | \( 1 - 5.05T + 13T^{2} \) |
| 17 | \( 1 - 5.31T + 17T^{2} \) |
| 19 | \( 1 - 2.25T + 19T^{2} \) |
| 23 | \( 1 - 1.05T + 23T^{2} \) |
| 29 | \( 1 - 2.79T + 29T^{2} \) |
| 31 | \( 1 - 3.74T + 31T^{2} \) |
| 37 | \( 1 + 0.791T + 37T^{2} \) |
| 41 | \( 1 + 6.15T + 41T^{2} \) |
| 43 | \( 1 + 2.70T + 43T^{2} \) |
| 47 | \( 1 + 11.2T + 47T^{2} \) |
| 53 | \( 1 - 1.05T + 53T^{2} \) |
| 59 | \( 1 - 4.53T + 59T^{2} \) |
| 61 | \( 1 - 9.88T + 61T^{2} \) |
| 67 | \( 1 - 10.4T + 67T^{2} \) |
| 71 | \( 1 + 1.05T + 71T^{2} \) |
| 73 | \( 1 + 10.1T + 73T^{2} \) |
| 79 | \( 1 + 9.41T + 79T^{2} \) |
| 83 | \( 1 + 3.15T + 83T^{2} \) |
| 89 | \( 1 + 12.5T + 89T^{2} \) |
| 97 | \( 1 + 17.7T + 97T^{2} \) |
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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
−11.16122765840668776504578353715, −10.07300450912709093210650386838, −8.526126637223041422063928618454, −8.198069418547224279393343611008, −6.77004672104555560321448485387, −5.64868869093569853807141647585, −5.18892538462457448070263611740, −4.31893288265050685272664887208, −3.22062470698399926968993478902, −1.16452410697109785082774396564,
1.16452410697109785082774396564, 3.22062470698399926968993478902, 4.31893288265050685272664887208, 5.18892538462457448070263611740, 5.64868869093569853807141647585, 6.77004672104555560321448485387, 8.198069418547224279393343611008, 8.526126637223041422063928618454, 10.07300450912709093210650386838, 11.16122765840668776504578353715
