| L(s) = 1 | − 3-s − 7-s + 9-s + 7·13-s + 4·17-s + 3·19-s + 21-s + 6·23-s − 27-s + 8·29-s + 3·31-s + 11·37-s − 7·39-s + 6·41-s − 43-s − 4·47-s + 49-s − 4·51-s + 6·53-s − 3·57-s − 6·59-s + 13·61-s − 63-s − 5·67-s − 6·69-s + 6·71-s + 3·73-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.377·7-s + 1/3·9-s + 1.94·13-s + 0.970·17-s + 0.688·19-s + 0.218·21-s + 1.25·23-s − 0.192·27-s + 1.48·29-s + 0.538·31-s + 1.80·37-s − 1.12·39-s + 0.937·41-s − 0.152·43-s − 0.583·47-s + 1/7·49-s − 0.560·51-s + 0.824·53-s − 0.397·57-s − 0.781·59-s + 1.66·61-s − 0.125·63-s − 0.610·67-s − 0.722·69-s + 0.712·71-s + 0.351·73-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
| \(L(1)\) |
\(\approx\) |
\(3.790579320\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.790579320\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 7 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 3 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 - 3 T + p T^{2} \) |
| 37 | \( 1 - 11 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 13 T + p T^{2} \) |
| 67 | \( 1 + 5 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 3 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−12.87353983934061, −12.35550142889849, −11.88818992040843, −11.35287724388029, −11.11160457066236, −10.55762990125003, −10.15070499059207, −9.561692959301454, −9.251236407578325, −8.572371922680549, −8.157637291718646, −7.745356870890954, −7.031150898996594, −6.572288360199842, −6.232798993839251, −5.645476801032525, −5.335817904295746, −4.601842717603333, −4.119502935221520, −3.550692622554757, −2.999543349321407, −2.577415587910774, −1.487447805186282, −0.9595333913575160, −0.7369596774002616,
0.7369596774002616, 0.9595333913575160, 1.487447805186282, 2.577415587910774, 2.999543349321407, 3.550692622554757, 4.119502935221520, 4.601842717603333, 5.335817904295746, 5.645476801032525, 6.232798993839251, 6.572288360199842, 7.031150898996594, 7.745356870890954, 8.157637291718646, 8.572371922680549, 9.251236407578325, 9.561692959301454, 10.15070499059207, 10.55762990125003, 11.11160457066236, 11.35287724388029, 11.88818992040843, 12.35550142889849, 12.87353983934061
