| L(s) = 1 | + 3·3-s + 2·7-s + 6·9-s − 13-s + 6·21-s − 23-s + 9·27-s − 3·29-s + 3·31-s + 8·37-s − 3·39-s + 3·41-s + 2·43-s + 11·47-s − 3·49-s + 14·53-s − 8·59-s − 4·61-s + 12·63-s + 4·67-s − 3·69-s + 7·71-s + 9·73-s + 9·81-s − 4·83-s − 9·87-s − 2·89-s + ⋯ |
| L(s) = 1 | + 1.73·3-s + 0.755·7-s + 2·9-s − 0.277·13-s + 1.30·21-s − 0.208·23-s + 1.73·27-s − 0.557·29-s + 0.538·31-s + 1.31·37-s − 0.480·39-s + 0.468·41-s + 0.304·43-s + 1.60·47-s − 3/7·49-s + 1.92·53-s − 1.04·59-s − 0.512·61-s + 1.51·63-s + 0.488·67-s − 0.361·69-s + 0.830·71-s + 1.05·73-s + 81-s − 0.439·83-s − 0.964·87-s − 0.211·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.316143919\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.316143919\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 3 | \( 1 - p T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 - 3 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 - 11 T + p T^{2} \) |
| 53 | \( 1 - 14 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 7 T + p T^{2} \) |
| 73 | \( 1 - 9 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
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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
−8.285574047889435899285775099263, −7.72080508917808983337092272200, −7.27420693336935451926548382562, −6.24709568035458347850164186333, −5.21601831815009121569251901191, −4.30937835957950367596720771655, −3.78023121707294719609388975674, −2.71063570111023727220455874686, −2.20807718970593446330580525082, −1.14993359895366307790634424705,
1.14993359895366307790634424705, 2.20807718970593446330580525082, 2.71063570111023727220455874686, 3.78023121707294719609388975674, 4.30937835957950367596720771655, 5.21601831815009121569251901191, 6.24709568035458347850164186333, 7.27420693336935451926548382562, 7.72080508917808983337092272200, 8.285574047889435899285775099263
