| L(s) = 1 | + 2.80·2-s − 2.60·3-s + 5.84·4-s − 7.30·6-s + 1.33·7-s + 10.7·8-s + 3.80·9-s + 1.09·11-s − 15.2·12-s − 3.17·13-s + 3.73·14-s + 18.4·16-s − 17-s + 10.6·18-s + 2.75·19-s − 3.47·21-s + 3.06·22-s − 3.57·23-s − 28.0·24-s − 8.88·26-s − 2.08·27-s + 7.79·28-s − 0.180·29-s + 0.816·31-s + 30.2·32-s − 2.85·33-s − 2.80·34-s + ⋯ |
| L(s) = 1 | + 1.98·2-s − 1.50·3-s + 2.92·4-s − 2.98·6-s + 0.503·7-s + 3.80·8-s + 1.26·9-s + 0.330·11-s − 4.40·12-s − 0.879·13-s + 0.997·14-s + 4.62·16-s − 0.242·17-s + 2.50·18-s + 0.632·19-s − 0.758·21-s + 0.653·22-s − 0.744·23-s − 5.73·24-s − 1.74·26-s − 0.402·27-s + 1.47·28-s − 0.0334·29-s + 0.146·31-s + 5.34·32-s − 0.497·33-s − 0.480·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.067902005\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.067902005\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 2 | \( 1 - 2.80T + 2T^{2} \) |
| 3 | \( 1 + 2.60T + 3T^{2} \) |
| 7 | \( 1 - 1.33T + 7T^{2} \) |
| 11 | \( 1 - 1.09T + 11T^{2} \) |
| 13 | \( 1 + 3.17T + 13T^{2} \) |
| 19 | \( 1 - 2.75T + 19T^{2} \) |
| 23 | \( 1 + 3.57T + 23T^{2} \) |
| 29 | \( 1 + 0.180T + 29T^{2} \) |
| 31 | \( 1 - 0.816T + 31T^{2} \) |
| 37 | \( 1 + 8.44T + 37T^{2} \) |
| 41 | \( 1 + 7.97T + 41T^{2} \) |
| 43 | \( 1 + 6.54T + 43T^{2} \) |
| 47 | \( 1 + 0.576T + 47T^{2} \) |
| 53 | \( 1 + 7.84T + 53T^{2} \) |
| 59 | \( 1 + 9.76T + 59T^{2} \) |
| 61 | \( 1 + 5.21T + 61T^{2} \) |
| 67 | \( 1 - 2.64T + 67T^{2} \) |
| 71 | \( 1 - 13.4T + 71T^{2} \) |
| 73 | \( 1 - 12.3T + 73T^{2} \) |
| 79 | \( 1 - 3.74T + 79T^{2} \) |
| 83 | \( 1 - 4.66T + 83T^{2} \) |
| 89 | \( 1 - 3.00T + 89T^{2} \) |
| 97 | \( 1 - 12.2T + 97T^{2} \) |
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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
−11.58618034640729666365271014166, −10.83648070452996643734124526880, −10.00783685752194775157529160633, −7.83117934323266389450833113356, −6.82287342654208620961233411325, −6.19824789833829410697511042837, −5.08260335648645955696449783511, −4.83713158263332715692091285772, −3.48382147393427179616127341770, −1.81067540079271052228576889822,
1.81067540079271052228576889822, 3.48382147393427179616127341770, 4.83713158263332715692091285772, 5.08260335648645955696449783511, 6.19824789833829410697511042837, 6.82287342654208620961233411325, 7.83117934323266389450833113356, 10.00783685752194775157529160633, 10.83648070452996643734124526880, 11.58618034640729666365271014166
