L(s) = 1 | + 2-s + 3-s + 4-s + 3.09·5-s + 6-s − 4.21·7-s + 8-s + 9-s + 3.09·10-s + 11-s + 12-s − 0.518·13-s − 4.21·14-s + 3.09·15-s + 16-s − 3.01·17-s + 18-s + 8.39·19-s + 3.09·20-s − 4.21·21-s + 22-s − 0.337·23-s + 24-s + 4.55·25-s − 0.518·26-s + 27-s − 4.21·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s + 1.38·5-s + 0.408·6-s − 1.59·7-s + 0.353·8-s + 0.333·9-s + 0.977·10-s + 0.301·11-s + 0.288·12-s − 0.143·13-s − 1.12·14-s + 0.798·15-s + 0.250·16-s − 0.730·17-s + 0.235·18-s + 1.92·19-s + 0.691·20-s − 0.919·21-s + 0.213·22-s − 0.0703·23-s + 0.204·24-s + 0.911·25-s − 0.101·26-s + 0.192·27-s − 0.796·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.446082977\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.446082977\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 61 | \( 1 + T \) |
good | 5 | \( 1 - 3.09T + 5T^{2} \) |
| 7 | \( 1 + 4.21T + 7T^{2} \) |
| 13 | \( 1 + 0.518T + 13T^{2} \) |
| 17 | \( 1 + 3.01T + 17T^{2} \) |
| 19 | \( 1 - 8.39T + 19T^{2} \) |
| 23 | \( 1 + 0.337T + 23T^{2} \) |
| 29 | \( 1 - 5.57T + 29T^{2} \) |
| 31 | \( 1 - 4.75T + 31T^{2} \) |
| 37 | \( 1 - 0.867T + 37T^{2} \) |
| 41 | \( 1 - 8.17T + 41T^{2} \) |
| 43 | \( 1 + 9.01T + 43T^{2} \) |
| 47 | \( 1 - 2.07T + 47T^{2} \) |
| 53 | \( 1 - 4.08T + 53T^{2} \) |
| 59 | \( 1 + 2.99T + 59T^{2} \) |
| 67 | \( 1 + 1.16T + 67T^{2} \) |
| 71 | \( 1 - 15.3T + 71T^{2} \) |
| 73 | \( 1 - 8.24T + 73T^{2} \) |
| 79 | \( 1 + 15.3T + 79T^{2} \) |
| 83 | \( 1 + 8.21T + 83T^{2} \) |
| 89 | \( 1 - 8.31T + 89T^{2} \) |
| 97 | \( 1 + 4.53T + 97T^{2} \) |
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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
−8.597599501659499016785429075618, −7.47094004992479715302610745211, −6.72523634874453392511499052199, −6.26564717803532544761952678882, −5.56575227144879787694298975973, −4.70908957299570958644085623952, −3.64742825584772285290382933898, −2.92208674387423020236062806379, −2.35341892986409573582347032842, −1.11249619646599480697799202311,
1.11249619646599480697799202311, 2.35341892986409573582347032842, 2.92208674387423020236062806379, 3.64742825584772285290382933898, 4.70908957299570958644085623952, 5.56575227144879787694298975973, 6.26564717803532544761952678882, 6.72523634874453392511499052199, 7.47094004992479715302610745211, 8.597599501659499016785429075618