| L(s) = 1 | + 4-s + 1.41·5-s − 1.41·7-s − 9-s − 1.41·11-s − 13-s + 16-s + 1.41·19-s + 1.41·20-s − 23-s + 1.00·25-s − 1.41·28-s − 2.00·35-s − 36-s + 1.41·37-s − 1.41·44-s − 1.41·45-s + 1.00·49-s − 52-s − 2.00·55-s + 1.41·63-s + 64-s − 1.41·65-s + 1.41·67-s + 1.41·76-s + 2.00·77-s + 1.41·80-s + ⋯ |
| L(s) = 1 | + 4-s + 1.41·5-s − 1.41·7-s − 9-s − 1.41·11-s − 13-s + 16-s + 1.41·19-s + 1.41·20-s − 23-s + 1.00·25-s − 1.41·28-s − 2.00·35-s − 36-s + 1.41·37-s − 1.41·44-s − 1.41·45-s + 1.00·49-s − 52-s − 2.00·55-s + 1.41·63-s + 64-s − 1.41·65-s + 1.41·67-s + 1.41·76-s + 2.00·77-s + 1.41·80-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 299 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 299 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9102674026\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9102674026\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| good | 2 | \( 1 - T^{2} \) |
| 3 | \( 1 + T^{2} \) |
| 5 | \( 1 - 1.41T + T^{2} \) |
| 7 | \( 1 + 1.41T + T^{2} \) |
| 11 | \( 1 + 1.41T + T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 - 1.41T + T^{2} \) |
| 29 | \( 1 + T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 - 1.41T + T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 - 1.41T + T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 + 1.41T + T^{2} \) |
| 89 | \( 1 + 1.41T + T^{2} \) |
| 97 | \( 1 + 1.41T + T^{2} \) |
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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
−12.05037906991414502063856336571, −10.92202813812348685629924878021, −9.831468177682174612584464385642, −9.703177125085313602866155489775, −8.029565257553100577928572433957, −6.96213253855295707899523009606, −5.91430280075578607857185415626, −5.45564159167117123449594132671, −2.99669904615245551410436173368, −2.41879826365451853934578904914,
2.41879826365451853934578904914, 2.99669904615245551410436173368, 5.45564159167117123449594132671, 5.91430280075578607857185415626, 6.96213253855295707899523009606, 8.029565257553100577928572433957, 9.703177125085313602866155489775, 9.831468177682174612584464385642, 10.92202813812348685629924878021, 12.05037906991414502063856336571
