| L(s) = 1 | + 2.74·2-s − 3-s + 5.53·4-s + 1.91·5-s − 2.74·6-s + 9.71·8-s + 9-s + 5.25·10-s + 5.32·11-s − 5.53·12-s + 4.78·13-s − 1.91·15-s + 15.6·16-s − 2.61·17-s + 2.74·18-s − 4.46·19-s + 10.6·20-s + 14.6·22-s − 23-s − 9.71·24-s − 1.33·25-s + 13.1·26-s − 27-s − 8.86·29-s − 5.25·30-s − 5.90·31-s + 23.4·32-s + ⋯ |
| L(s) = 1 | + 1.94·2-s − 0.577·3-s + 2.76·4-s + 0.856·5-s − 1.12·6-s + 3.43·8-s + 0.333·9-s + 1.66·10-s + 1.60·11-s − 1.59·12-s + 1.32·13-s − 0.494·15-s + 3.90·16-s − 0.634·17-s + 0.647·18-s − 1.02·19-s + 2.37·20-s + 3.11·22-s − 0.208·23-s − 1.98·24-s − 0.266·25-s + 2.57·26-s − 0.192·27-s − 1.64·29-s − 0.959·30-s − 1.06·31-s + 4.14·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.415598542\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.415598542\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 7 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 2 | \( 1 - 2.74T + 2T^{2} \) |
| 5 | \( 1 - 1.91T + 5T^{2} \) |
| 11 | \( 1 - 5.32T + 11T^{2} \) |
| 13 | \( 1 - 4.78T + 13T^{2} \) |
| 17 | \( 1 + 2.61T + 17T^{2} \) |
| 19 | \( 1 + 4.46T + 19T^{2} \) |
| 29 | \( 1 + 8.86T + 29T^{2} \) |
| 31 | \( 1 + 5.90T + 31T^{2} \) |
| 37 | \( 1 + 2.91T + 37T^{2} \) |
| 41 | \( 1 + 4.98T + 41T^{2} \) |
| 43 | \( 1 + 6.45T + 43T^{2} \) |
| 47 | \( 1 - 6.94T + 47T^{2} \) |
| 53 | \( 1 - 1.01T + 53T^{2} \) |
| 59 | \( 1 + 6.15T + 59T^{2} \) |
| 61 | \( 1 + 2.24T + 61T^{2} \) |
| 67 | \( 1 - 9.38T + 67T^{2} \) |
| 71 | \( 1 + 10.2T + 71T^{2} \) |
| 73 | \( 1 + 3.85T + 73T^{2} \) |
| 79 | \( 1 - 2.97T + 79T^{2} \) |
| 83 | \( 1 - 2.29T + 83T^{2} \) |
| 89 | \( 1 - 10.0T + 89T^{2} \) |
| 97 | \( 1 + 13.2T + 97T^{2} \) |
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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
−8.598079414045278436190674292790, −7.29922671840588669582714916108, −6.65301113637903121452295175847, −6.06308300638876302978008938700, −5.76002419202505016100025677835, −4.77278643620350393190972416225, −3.92187112670796733866935240082, −3.54510105249804347902993486745, −2.01015861744366119890084284510, −1.58424443317299372743401366327,
1.58424443317299372743401366327, 2.01015861744366119890084284510, 3.54510105249804347902993486745, 3.92187112670796733866935240082, 4.77278643620350393190972416225, 5.76002419202505016100025677835, 6.06308300638876302978008938700, 6.65301113637903121452295175847, 7.29922671840588669582714916108, 8.598079414045278436190674292790
