| L(s) = 1 | + 2·2-s + 2·4-s + 3·5-s − 4·7-s − 3·9-s + 6·10-s + 4·11-s − 13-s − 8·14-s − 4·16-s + 17-s − 6·18-s − 3·19-s + 6·20-s + 8·22-s + 6·23-s + 4·25-s − 2·26-s − 8·28-s + 3·29-s − 8·31-s − 8·32-s + 2·34-s − 12·35-s − 6·36-s + 2·37-s − 6·38-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 4-s + 1.34·5-s − 1.51·7-s − 9-s + 1.89·10-s + 1.20·11-s − 0.277·13-s − 2.13·14-s − 16-s + 0.242·17-s − 1.41·18-s − 0.688·19-s + 1.34·20-s + 1.70·22-s + 1.25·23-s + 4/5·25-s − 0.392·26-s − 1.51·28-s + 0.557·29-s − 1.43·31-s − 1.41·32-s + 0.342·34-s − 2.02·35-s − 36-s + 0.328·37-s − 0.973·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 179 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 179 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.260198254\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.260198254\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 179 | \( 1 - T \) |
| good | 2 | \( 1 - p T + p T^{2} \) |
| 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 - 3 T + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 + 3 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 + 11 T + p T^{2} \) |
| 47 | \( 1 - T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + 5 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 + 9 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - 17 T + p T^{2} \) |
| 89 | \( 1 + T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−12.91966077992607962496402669630, −12.11851077920995702575878930036, −10.91540134714576928534303180516, −9.527249149175652297561485725769, −9.039531184742746287388806553536, −6.69315229163834238910355551443, −6.18510863780788689281989494770, −5.26927125289354544444142895540, −3.65455224933898053965548391576, −2.57742433461389850447314956434,
2.57742433461389850447314956434, 3.65455224933898053965548391576, 5.26927125289354544444142895540, 6.18510863780788689281989494770, 6.69315229163834238910355551443, 9.039531184742746287388806553536, 9.527249149175652297561485725769, 10.91540134714576928534303180516, 12.11851077920995702575878930036, 12.91966077992607962496402669630
