| L(s) = 1 | − 4·7-s − 2·11-s + 4·13-s − 17-s − 4·19-s + 8·23-s − 8·29-s − 2·37-s + 4·41-s + 6·43-s + 12·47-s + 9·49-s − 14·53-s + 2·61-s + 2·67-s − 14·71-s + 2·73-s + 8·77-s + 4·79-s + 16·83-s + 6·89-s − 16·91-s − 14·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
| L(s) = 1 | − 1.51·7-s − 0.603·11-s + 1.10·13-s − 0.242·17-s − 0.917·19-s + 1.66·23-s − 1.48·29-s − 0.328·37-s + 0.624·41-s + 0.914·43-s + 1.75·47-s + 9/7·49-s − 1.92·53-s + 0.256·61-s + 0.244·67-s − 1.66·71-s + 0.234·73-s + 0.911·77-s + 0.450·79-s + 1.75·83-s + 0.635·89-s − 1.67·91-s − 1.42·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 30600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 30600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.160650575\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.160650575\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 4 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 14 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + 14 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−15.21719507152081, −14.70495737165414, −13.91743340589349, −13.32226935692639, −13.02564694702258, −12.68677827467821, −12.06155936988781, −11.17077479974347, −10.73480351221770, −10.52381762493881, −9.524827582299621, −9.181181207024080, −8.822014795744056, −7.953096083719705, −7.397854566532122, −6.709994574332086, −6.298616126498418, −5.709114353596225, −5.102234008539773, −4.135974643619337, −3.706567338440152, −2.959075782784350, −2.453373720139545, −1.391345428321141, −0.4199784970204486,
0.4199784970204486, 1.391345428321141, 2.453373720139545, 2.959075782784350, 3.706567338440152, 4.135974643619337, 5.102234008539773, 5.709114353596225, 6.298616126498418, 6.709994574332086, 7.397854566532122, 7.953096083719705, 8.822014795744056, 9.181181207024080, 9.524827582299621, 10.52381762493881, 10.73480351221770, 11.17077479974347, 12.06155936988781, 12.68677827467821, 13.02564694702258, 13.32226935692639, 13.91743340589349, 14.70495737165414, 15.21719507152081
