L(s) = 1 | − 2-s − 3-s + 4-s − 0.937·5-s + 6-s − 0.792·7-s − 8-s + 9-s + 0.937·10-s − 4.70·11-s − 12-s + 13-s + 0.792·14-s + 0.937·15-s + 16-s − 3.60·17-s − 18-s − 0.803·19-s − 0.937·20-s + 0.792·21-s + 4.70·22-s − 4.68·23-s + 24-s − 4.12·25-s − 26-s − 27-s − 0.792·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.419·5-s + 0.408·6-s − 0.299·7-s − 0.353·8-s + 0.333·9-s + 0.296·10-s − 1.41·11-s − 0.288·12-s + 0.277·13-s + 0.211·14-s + 0.241·15-s + 0.250·16-s − 0.874·17-s − 0.235·18-s − 0.184·19-s − 0.209·20-s + 0.172·21-s + 1.00·22-s − 0.977·23-s + 0.204·24-s − 0.824·25-s − 0.196·26-s − 0.192·27-s − 0.149·28-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
\(L(1)\) |
\(\approx\) |
\(0.1745776687\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1745776687\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + 0.937T + 5T^{2} \) |
| 7 | \( 1 + 0.792T + 7T^{2} \) |
| 11 | \( 1 + 4.70T + 11T^{2} \) |
| 17 | \( 1 + 3.60T + 17T^{2} \) |
| 19 | \( 1 + 0.803T + 19T^{2} \) |
| 23 | \( 1 + 4.68T + 23T^{2} \) |
| 29 | \( 1 + 3.11T + 29T^{2} \) |
| 31 | \( 1 + 3.57T + 31T^{2} \) |
| 37 | \( 1 - 1.63T + 37T^{2} \) |
| 41 | \( 1 + 7.48T + 41T^{2} \) |
| 43 | \( 1 - 3.98T + 43T^{2} \) |
| 47 | \( 1 - 0.579T + 47T^{2} \) |
| 53 | \( 1 - 5.61T + 53T^{2} \) |
| 59 | \( 1 + 1.93T + 59T^{2} \) |
| 61 | \( 1 + 14.8T + 61T^{2} \) |
| 67 | \( 1 - 5.77T + 67T^{2} \) |
| 71 | \( 1 + 5.96T + 71T^{2} \) |
| 73 | \( 1 - 0.810T + 73T^{2} \) |
| 79 | \( 1 - 6.43T + 79T^{2} \) |
| 83 | \( 1 + 5.41T + 83T^{2} \) |
| 89 | \( 1 + 18.2T + 89T^{2} \) |
| 97 | \( 1 - 10.4T + 97T^{2} \) |
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
−7.73275971890809641383897357655, −7.36679241956815382441927222551, −6.44088307217929729662718700831, −5.89814471742508835090885321558, −5.15944562936389920536260832967, −4.29218561355973869065711759080, −3.48174450359037810192729394030, −2.48638335400858269477152675958, −1.69069598273965836967808354039, −0.22739236479326189977802727441,
0.22739236479326189977802727441, 1.69069598273965836967808354039, 2.48638335400858269477152675958, 3.48174450359037810192729394030, 4.29218561355973869065711759080, 5.15944562936389920536260832967, 5.89814471742508835090885321558, 6.44088307217929729662718700831, 7.36679241956815382441927222551, 7.73275971890809641383897357655