| L(s) = 1 | − 2.82·3-s + 5.00·9-s + 4·11-s + 4.24·13-s + 1.41·17-s − 2.82·19-s + 4·23-s − 5.65·27-s + 8·29-s − 11.3·33-s + 8·37-s − 12·39-s + 7.07·41-s + 4·43-s + 5.65·47-s − 4.00·51-s − 10·53-s + 8.00·57-s − 14.1·59-s + 7.07·61-s − 11.3·69-s − 7.07·73-s + 8·79-s + 1.00·81-s − 14.1·83-s − 22.6·87-s − 7.07·89-s + ⋯ |
| L(s) = 1 | − 1.63·3-s + 1.66·9-s + 1.20·11-s + 1.17·13-s + 0.342·17-s − 0.648·19-s + 0.834·23-s − 1.08·27-s + 1.48·29-s − 1.96·33-s + 1.31·37-s − 1.92·39-s + 1.10·41-s + 0.609·43-s + 0.825·47-s − 0.560·51-s − 1.37·53-s + 1.05·57-s − 1.84·59-s + 0.905·61-s − 1.36·69-s − 0.827·73-s + 0.900·79-s + 0.111·81-s − 1.55·83-s − 2.42·87-s − 0.749·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.359233941\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.359233941\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + 2.82T + 3T^{2} \) |
| 11 | \( 1 - 4T + 11T^{2} \) |
| 13 | \( 1 - 4.24T + 13T^{2} \) |
| 17 | \( 1 - 1.41T + 17T^{2} \) |
| 19 | \( 1 + 2.82T + 19T^{2} \) |
| 23 | \( 1 - 4T + 23T^{2} \) |
| 29 | \( 1 - 8T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 8T + 37T^{2} \) |
| 41 | \( 1 - 7.07T + 41T^{2} \) |
| 43 | \( 1 - 4T + 43T^{2} \) |
| 47 | \( 1 - 5.65T + 47T^{2} \) |
| 53 | \( 1 + 10T + 53T^{2} \) |
| 59 | \( 1 + 14.1T + 59T^{2} \) |
| 61 | \( 1 - 7.07T + 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 7.07T + 73T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 + 14.1T + 83T^{2} \) |
| 89 | \( 1 + 7.07T + 89T^{2} \) |
| 97 | \( 1 + 1.41T + 97T^{2} \) |
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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
−8.254827282833983292048350942376, −7.30331539016476462806141114305, −6.45831871903475564891624811743, −6.22615383318520579364249974873, −5.51997141698656966784947398309, −4.50796088206760797707562016831, −4.12581084462365198197487946089, −2.93718357607812468484050016290, −1.41474289686561157280179280886, −0.798854503574070641871186099960,
0.798854503574070641871186099960, 1.41474289686561157280179280886, 2.93718357607812468484050016290, 4.12581084462365198197487946089, 4.50796088206760797707562016831, 5.51997141698656966784947398309, 6.22615383318520579364249974873, 6.45831871903475564891624811743, 7.30331539016476462806141114305, 8.254827282833983292048350942376
