| L(s) = 1 | − 1.41·2-s + 1.41·3-s + 4.41·5-s − 2.00·6-s + 2.82·8-s − 0.999·9-s − 6.24·10-s + 4.24·11-s + 6.24·15-s − 4.00·16-s + 1.41·17-s + 1.41·18-s + 1.24·19-s − 6·22-s − 0.171·23-s + 4·24-s + 14.4·25-s − 5.65·27-s + 5.82·29-s − 8.82·30-s − 5.24·31-s + 6·33-s − 2.00·34-s + 6.24·37-s + ⋯ |
| L(s) = 1 | − 1.00·2-s + 0.816·3-s + 1.97·5-s − 0.816·6-s + 0.999·8-s − 0.333·9-s − 1.97·10-s + 1.27·11-s + 1.61·15-s − 1.00·16-s + 0.342·17-s + 0.333·18-s + 0.285·19-s − 1.27·22-s − 0.0357·23-s + 0.816·24-s + 2.89·25-s − 1.08·27-s + 1.08·29-s − 1.61·30-s − 0.941·31-s + 1.04·33-s − 0.342·34-s + 1.02·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8281 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8281 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.586480524\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.586480524\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 1.41T + 2T^{2} \) |
| 3 | \( 1 - 1.41T + 3T^{2} \) |
| 5 | \( 1 - 4.41T + 5T^{2} \) |
| 11 | \( 1 - 4.24T + 11T^{2} \) |
| 17 | \( 1 - 1.41T + 17T^{2} \) |
| 19 | \( 1 - 1.24T + 19T^{2} \) |
| 23 | \( 1 + 0.171T + 23T^{2} \) |
| 29 | \( 1 - 5.82T + 29T^{2} \) |
| 31 | \( 1 + 5.24T + 31T^{2} \) |
| 37 | \( 1 - 6.24T + 37T^{2} \) |
| 41 | \( 1 - 3.17T + 41T^{2} \) |
| 43 | \( 1 + 5T + 43T^{2} \) |
| 47 | \( 1 - 4.41T + 47T^{2} \) |
| 53 | \( 1 + 5.82T + 53T^{2} \) |
| 59 | \( 1 - 11.6T + 59T^{2} \) |
| 61 | \( 1 + 6T + 61T^{2} \) |
| 67 | \( 1 + 2.48T + 67T^{2} \) |
| 71 | \( 1 + 1.07T + 71T^{2} \) |
| 73 | \( 1 + 0.757T + 73T^{2} \) |
| 79 | \( 1 + 1.48T + 79T^{2} \) |
| 83 | \( 1 - 4.75T + 83T^{2} \) |
| 89 | \( 1 - 4.41T + 89T^{2} \) |
| 97 | \( 1 + 13.7T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.084706058811557247177113643437, −7.22031304022754519184931302990, −6.50518779953623242458192190980, −5.86207733477538719169145359748, −5.14421431012044723434949267705, −4.24286326295025153714880573455, −3.22410324106254068682997310775, −2.37843331726855602961279424683, −1.66725772256451209926018586536, −0.966202428933816318086328849034,
0.966202428933816318086328849034, 1.66725772256451209926018586536, 2.37843331726855602961279424683, 3.22410324106254068682997310775, 4.24286326295025153714880573455, 5.14421431012044723434949267705, 5.86207733477538719169145359748, 6.50518779953623242458192190980, 7.22031304022754519184931302990, 8.084706058811557247177113643437
