| L(s) = 1 | − 3·3-s + 5·7-s + 6·9-s + 4·13-s − 17-s + 5·19-s − 15·21-s + 4·23-s − 9·27-s + 5·29-s + 31-s − 3·37-s − 12·39-s + 2·41-s + 8·43-s + 6·47-s + 18·49-s + 3·51-s − 11·53-s − 15·57-s − 5·61-s + 30·63-s − 4·67-s − 12·69-s + 15·71-s − 6·73-s − 8·79-s + ⋯ |
| L(s) = 1 | − 1.73·3-s + 1.88·7-s + 2·9-s + 1.10·13-s − 0.242·17-s + 1.14·19-s − 3.27·21-s + 0.834·23-s − 1.73·27-s + 0.928·29-s + 0.179·31-s − 0.493·37-s − 1.92·39-s + 0.312·41-s + 1.21·43-s + 0.875·47-s + 18/7·49-s + 0.420·51-s − 1.51·53-s − 1.98·57-s − 0.640·61-s + 3.77·63-s − 0.488·67-s − 1.44·69-s + 1.78·71-s − 0.702·73-s − 0.900·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 48400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.321797374\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.321797374\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 + p T + p T^{2} \) |
| 7 | \( 1 - 5 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 5 T + p T^{2} \) |
| 31 | \( 1 - T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 11 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 5 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 15 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 - T + p T^{2} \) |
| 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
−14.48040442122394, −14.01605188131615, −13.59498122907696, −12.87928292792179, −12.22109221359115, −11.95121167234206, −11.33109541279235, −10.97839812398464, −10.80558197369598, −10.14884185883597, −9.370928329557599, −8.749237617758032, −8.194728179623487, −7.491822927865101, −7.186920568335677, −6.322038796337954, −5.944972169724051, −5.265128024316001, −4.931265462700208, −4.436134522617854, −3.792005499695722, −2.777842491426451, −1.740088444488470, −1.176491017820456, −0.7297321593050434,
0.7297321593050434, 1.176491017820456, 1.740088444488470, 2.777842491426451, 3.792005499695722, 4.436134522617854, 4.931265462700208, 5.265128024316001, 5.944972169724051, 6.322038796337954, 7.186920568335677, 7.491822927865101, 8.194728179623487, 8.749237617758032, 9.370928329557599, 10.14884185883597, 10.80558197369598, 10.97839812398464, 11.33109541279235, 11.95121167234206, 12.22109221359115, 12.87928292792179, 13.59498122907696, 14.01605188131615, 14.48040442122394
