| L(s) = 1 | − 2.74·2-s − 3-s + 5.56·4-s + 3.91·5-s + 2.74·6-s + 4.39·7-s − 9.79·8-s + 9-s − 10.7·10-s + 11-s − 5.56·12-s − 12.0·14-s − 3.91·15-s + 15.8·16-s + 2.57·17-s − 2.74·18-s + 3.52·19-s + 21.7·20-s − 4.39·21-s − 2.74·22-s + 2.82·23-s + 9.79·24-s + 10.2·25-s − 27-s + 24.4·28-s − 3.74·29-s + 10.7·30-s + ⋯ |
| L(s) = 1 | − 1.94·2-s − 0.577·3-s + 2.78·4-s + 1.74·5-s + 1.12·6-s + 1.65·7-s − 3.46·8-s + 0.333·9-s − 3.40·10-s + 0.301·11-s − 1.60·12-s − 3.22·14-s − 1.00·15-s + 3.95·16-s + 0.625·17-s − 0.648·18-s + 0.807·19-s + 4.86·20-s − 0.958·21-s − 0.586·22-s + 0.588·23-s + 1.99·24-s + 2.05·25-s − 0.192·27-s + 4.61·28-s − 0.695·29-s + 1.96·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5577 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5577 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.385412603\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.385412603\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 2.74T + 2T^{2} \) |
| 5 | \( 1 - 3.91T + 5T^{2} \) |
| 7 | \( 1 - 4.39T + 7T^{2} \) |
| 17 | \( 1 - 2.57T + 17T^{2} \) |
| 19 | \( 1 - 3.52T + 19T^{2} \) |
| 23 | \( 1 - 2.82T + 23T^{2} \) |
| 29 | \( 1 + 3.74T + 29T^{2} \) |
| 31 | \( 1 - 0.391T + 31T^{2} \) |
| 37 | \( 1 + 3.93T + 37T^{2} \) |
| 41 | \( 1 + 7.05T + 41T^{2} \) |
| 43 | \( 1 + 2.21T + 43T^{2} \) |
| 47 | \( 1 + 5.14T + 47T^{2} \) |
| 53 | \( 1 - 11.0T + 53T^{2} \) |
| 59 | \( 1 + 15.1T + 59T^{2} \) |
| 61 | \( 1 - 2.50T + 61T^{2} \) |
| 67 | \( 1 - 0.577T + 67T^{2} \) |
| 71 | \( 1 + 2.69T + 71T^{2} \) |
| 73 | \( 1 - 0.329T + 73T^{2} \) |
| 79 | \( 1 - 2.89T + 79T^{2} \) |
| 83 | \( 1 - 0.425T + 83T^{2} \) |
| 89 | \( 1 - 16.0T + 89T^{2} \) |
| 97 | \( 1 - 9.38T + 97T^{2} \) |
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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
−8.274440970821589560063461445931, −7.53311447650598791142744499438, −6.93433900815120853319793318030, −6.16361088533062187357641191253, −5.49143878732362908876315624426, −4.96940287359232132176292239383, −3.21057373219967485750220545475, −2.06967564901007450914309458532, −1.61314006606349572451154014943, −0.961629595782257116230242393760,
0.961629595782257116230242393760, 1.61314006606349572451154014943, 2.06967564901007450914309458532, 3.21057373219967485750220545475, 4.96940287359232132176292239383, 5.49143878732362908876315624426, 6.16361088533062187357641191253, 6.93433900815120853319793318030, 7.53311447650598791142744499438, 8.274440970821589560063461445931
