L(s) = 1 | + 2-s + 3-s + 4-s + 0.197·5-s + 6-s + 1.50·7-s + 8-s + 9-s + 0.197·10-s − 1.06·11-s + 12-s − 3.56·13-s + 1.50·14-s + 0.197·15-s + 16-s + 17-s + 18-s − 3.08·19-s + 0.197·20-s + 1.50·21-s − 1.06·22-s + 2.60·23-s + 24-s − 4.96·25-s − 3.56·26-s + 27-s + 1.50·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.0881·5-s + 0.408·6-s + 0.570·7-s + 0.353·8-s + 0.333·9-s + 0.0623·10-s − 0.321·11-s + 0.288·12-s − 0.990·13-s + 0.403·14-s + 0.0509·15-s + 0.250·16-s + 0.242·17-s + 0.235·18-s − 0.708·19-s + 0.0440·20-s + 0.329·21-s − 0.227·22-s + 0.543·23-s + 0.204·24-s − 0.992·25-s − 0.700·26-s + 0.192·27-s + 0.285·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.329285425\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.329285425\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| 59 | \( 1 + T \) |
good | 5 | \( 1 - 0.197T + 5T^{2} \) |
| 7 | \( 1 - 1.50T + 7T^{2} \) |
| 11 | \( 1 + 1.06T + 11T^{2} \) |
| 13 | \( 1 + 3.56T + 13T^{2} \) |
| 19 | \( 1 + 3.08T + 19T^{2} \) |
| 23 | \( 1 - 2.60T + 23T^{2} \) |
| 29 | \( 1 - 1.69T + 29T^{2} \) |
| 31 | \( 1 - 8.35T + 31T^{2} \) |
| 37 | \( 1 - 10.5T + 37T^{2} \) |
| 41 | \( 1 - 10.3T + 41T^{2} \) |
| 43 | \( 1 - 3.49T + 43T^{2} \) |
| 47 | \( 1 - 8.36T + 47T^{2} \) |
| 53 | \( 1 + 9.48T + 53T^{2} \) |
| 61 | \( 1 - 13.8T + 61T^{2} \) |
| 67 | \( 1 - 13.1T + 67T^{2} \) |
| 71 | \( 1 - 0.742T + 71T^{2} \) |
| 73 | \( 1 + 7.50T + 73T^{2} \) |
| 79 | \( 1 - 10.8T + 79T^{2} \) |
| 83 | \( 1 + 1.77T + 83T^{2} \) |
| 89 | \( 1 - 4.46T + 89T^{2} \) |
| 97 | \( 1 - 15.6T + 97T^{2} \) |
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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
−7.78242522791784101922027237710, −7.62673964191489638856010070431, −6.55938641680148477425533820575, −5.94227967703476451280051565951, −4.98535579017595416983147896717, −4.50054578718650555860292491490, −3.73772637394168283187110924871, −2.57898862585321725688402916947, −2.32188230101146007011793051431, −0.975200664269059715152308219147,
0.975200664269059715152308219147, 2.32188230101146007011793051431, 2.57898862585321725688402916947, 3.73772637394168283187110924871, 4.50054578718650555860292491490, 4.98535579017595416983147896717, 5.94227967703476451280051565951, 6.55938641680148477425533820575, 7.62673964191489638856010070431, 7.78242522791784101922027237710