| L(s) = 1 | + 2-s + 3-s + 4-s − 3.73·5-s + 6-s − 3.81·7-s + 8-s + 9-s − 3.73·10-s − 2.78·11-s + 12-s + 13-s − 3.81·14-s − 3.73·15-s + 16-s − 1.82·17-s + 18-s − 4.81·19-s − 3.73·20-s − 3.81·21-s − 2.78·22-s − 7.96·23-s + 24-s + 8.91·25-s + 26-s + 27-s − 3.81·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s − 1.66·5-s + 0.408·6-s − 1.44·7-s + 0.353·8-s + 0.333·9-s − 1.17·10-s − 0.840·11-s + 0.288·12-s + 0.277·13-s − 1.02·14-s − 0.963·15-s + 0.250·16-s − 0.443·17-s + 0.235·18-s − 1.10·19-s − 0.834·20-s − 0.832·21-s − 0.594·22-s − 1.65·23-s + 0.204·24-s + 1.78·25-s + 0.196·26-s + 0.192·27-s − 0.721·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.233650926\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.233650926\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 103 | \( 1 + T \) |
| good | 5 | \( 1 + 3.73T + 5T^{2} \) |
| 7 | \( 1 + 3.81T + 7T^{2} \) |
| 11 | \( 1 + 2.78T + 11T^{2} \) |
| 17 | \( 1 + 1.82T + 17T^{2} \) |
| 19 | \( 1 + 4.81T + 19T^{2} \) |
| 23 | \( 1 + 7.96T + 23T^{2} \) |
| 29 | \( 1 - 1.52T + 29T^{2} \) |
| 31 | \( 1 - 4.81T + 31T^{2} \) |
| 37 | \( 1 + 2.06T + 37T^{2} \) |
| 41 | \( 1 - 9.25T + 41T^{2} \) |
| 43 | \( 1 + 12.4T + 43T^{2} \) |
| 47 | \( 1 - 7.09T + 47T^{2} \) |
| 53 | \( 1 - 8.32T + 53T^{2} \) |
| 59 | \( 1 + 11.8T + 59T^{2} \) |
| 61 | \( 1 - 7.08T + 61T^{2} \) |
| 67 | \( 1 - 0.370T + 67T^{2} \) |
| 71 | \( 1 + 5.96T + 71T^{2} \) |
| 73 | \( 1 + 8.16T + 73T^{2} \) |
| 79 | \( 1 + 0.128T + 79T^{2} \) |
| 83 | \( 1 + 0.0889T + 83T^{2} \) |
| 89 | \( 1 - 16.1T + 89T^{2} \) |
| 97 | \( 1 + 5.41T + 97T^{2} \) |
| 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
−7.76654526764952352039393499790, −7.17202617111436439337075101767, −6.45237665045150257628389277237, −5.89846346930938753718582523436, −4.67210561263348566315253413072, −4.15662054698844661269178880204, −3.56416105859041655070712481387, −2.95131884841937422572442023806, −2.17578498671790579823692030074, −0.44485692871559063761591385617,
0.44485692871559063761591385617, 2.17578498671790579823692030074, 2.95131884841937422572442023806, 3.56416105859041655070712481387, 4.15662054698844661269178880204, 4.67210561263348566315253413072, 5.89846346930938753718582523436, 6.45237665045150257628389277237, 7.17202617111436439337075101767, 7.76654526764952352039393499790
