| L(s) = 1 | + 2.58·2-s − 1.87·3-s + 4.67·4-s + 0.728·5-s − 4.83·6-s + 7-s + 6.90·8-s + 0.498·9-s + 1.88·10-s + 1.55·11-s − 8.74·12-s + 1.65·13-s + 2.58·14-s − 1.36·15-s + 8.49·16-s − 4.18·17-s + 1.28·18-s − 3.46·19-s + 3.40·20-s − 1.87·21-s + 4.02·22-s + 4.67·23-s − 12.9·24-s − 4.46·25-s + 4.28·26-s + 4.67·27-s + 4.67·28-s + ⋯ |
| L(s) = 1 | + 1.82·2-s − 1.07·3-s + 2.33·4-s + 0.325·5-s − 1.97·6-s + 0.377·7-s + 2.44·8-s + 0.166·9-s + 0.594·10-s + 0.469·11-s − 2.52·12-s + 0.460·13-s + 0.690·14-s − 0.351·15-s + 2.12·16-s − 1.01·17-s + 0.303·18-s − 0.794·19-s + 0.761·20-s − 0.408·21-s + 0.858·22-s + 0.973·23-s − 2.63·24-s − 0.893·25-s + 0.840·26-s + 0.900·27-s + 0.883·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 301 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 301 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.761509029\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.761509029\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 - T \) |
| 43 | \( 1 + T \) |
| good | 2 | \( 1 - 2.58T + 2T^{2} \) |
| 3 | \( 1 + 1.87T + 3T^{2} \) |
| 5 | \( 1 - 0.728T + 5T^{2} \) |
| 11 | \( 1 - 1.55T + 11T^{2} \) |
| 13 | \( 1 - 1.65T + 13T^{2} \) |
| 17 | \( 1 + 4.18T + 17T^{2} \) |
| 19 | \( 1 + 3.46T + 19T^{2} \) |
| 23 | \( 1 - 4.67T + 23T^{2} \) |
| 29 | \( 1 + 8.75T + 29T^{2} \) |
| 31 | \( 1 + 4.09T + 31T^{2} \) |
| 37 | \( 1 - 0.253T + 37T^{2} \) |
| 41 | \( 1 + 4.61T + 41T^{2} \) |
| 47 | \( 1 - 7.31T + 47T^{2} \) |
| 53 | \( 1 + 2.87T + 53T^{2} \) |
| 59 | \( 1 - 8.50T + 59T^{2} \) |
| 61 | \( 1 - 5.07T + 61T^{2} \) |
| 67 | \( 1 + 10.4T + 67T^{2} \) |
| 71 | \( 1 - 12.1T + 71T^{2} \) |
| 73 | \( 1 + 8.13T + 73T^{2} \) |
| 79 | \( 1 - 7.45T + 79T^{2} \) |
| 83 | \( 1 + 12.1T + 83T^{2} \) |
| 89 | \( 1 - 9.57T + 89T^{2} \) |
| 97 | \( 1 - 18.4T + 97T^{2} \) |
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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
−11.69579007612323337572542516205, −11.27613565608819634975396099202, −10.55862142014691830290214556293, −8.896808505937796521444577663404, −7.22663467130202243885793398326, −6.32289924919553707159105567112, −5.65161571917332461360873291550, −4.76275699173567542076742484592, −3.72017953175978764992954434935, −2.04138296252729884877785488820,
2.04138296252729884877785488820, 3.72017953175978764992954434935, 4.76275699173567542076742484592, 5.65161571917332461360873291550, 6.32289924919553707159105567112, 7.22663467130202243885793398326, 8.896808505937796521444577663404, 10.55862142014691830290214556293, 11.27613565608819634975396099202, 11.69579007612323337572542516205
