| L(s) = 1 | − 3-s + 7-s + 9-s + 2·13-s + 4·19-s − 21-s + 6·23-s − 27-s − 6·29-s + 8·31-s − 2·37-s − 2·39-s − 12·41-s − 4·43-s − 12·47-s + 49-s + 6·53-s − 4·57-s + 10·61-s + 63-s − 8·67-s − 6·69-s + 6·71-s − 10·73-s + 4·79-s + 81-s − 12·83-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.377·7-s + 1/3·9-s + 0.554·13-s + 0.917·19-s − 0.218·21-s + 1.25·23-s − 0.192·27-s − 1.11·29-s + 1.43·31-s − 0.328·37-s − 0.320·39-s − 1.87·41-s − 0.609·43-s − 1.75·47-s + 1/7·49-s + 0.824·53-s − 0.529·57-s + 1.28·61-s + 0.125·63-s − 0.977·67-s − 0.722·69-s + 0.712·71-s − 1.17·73-s + 0.450·79-s + 1/9·81-s − 1.31·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 254100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 254100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.933506646\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.933506646\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−12.99883997297236, −12.20816993079929, −11.80291564293265, −11.47281844107219, −11.17770848361638, −10.44653487979095, −10.20101499324805, −9.648867208563276, −9.156761358530377, −8.476198155530643, −8.338046719964454, −7.533912909298543, −7.194807267237750, −6.604240516194492, −6.275144748849828, −5.559170190722784, −5.067162793681911, −4.900714381089657, −4.116994438567178, −3.496316607400061, −3.125436499534300, −2.359186411779449, −1.548418323523227, −1.226108784042067, −0.4165591033312770,
0.4165591033312770, 1.226108784042067, 1.548418323523227, 2.359186411779449, 3.125436499534300, 3.496316607400061, 4.116994438567178, 4.900714381089657, 5.067162793681911, 5.559170190722784, 6.275144748849828, 6.604240516194492, 7.194807267237750, 7.533912909298543, 8.338046719964454, 8.476198155530643, 9.156761358530377, 9.648867208563276, 10.20101499324805, 10.44653487979095, 11.17770848361638, 11.47281844107219, 11.80291564293265, 12.20816993079929, 12.99883997297236
