L(s) = 1 | − 2-s + 3-s + 4-s + 2.63·5-s − 6-s + 3.87·7-s − 8-s + 9-s − 2.63·10-s + 3.26·11-s + 12-s + 2.63·13-s − 3.87·14-s + 2.63·15-s + 16-s − 17-s − 18-s + 3.97·19-s + 2.63·20-s + 3.87·21-s − 3.26·22-s + 2.46·23-s − 24-s + 1.92·25-s − 2.63·26-s + 27-s + 3.87·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s + 1.17·5-s − 0.408·6-s + 1.46·7-s − 0.353·8-s + 0.333·9-s − 0.831·10-s + 0.983·11-s + 0.288·12-s + 0.729·13-s − 1.03·14-s + 0.679·15-s + 0.250·16-s − 0.242·17-s − 0.235·18-s + 0.911·19-s + 0.588·20-s + 0.846·21-s − 0.695·22-s + 0.513·23-s − 0.204·24-s + 0.384·25-s − 0.515·26-s + 0.192·27-s + 0.733·28-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
\(L(1)\) |
\(\approx\) |
\(3.330784830\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.330784830\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - 2.63T + 5T^{2} \) |
| 7 | \( 1 - 3.87T + 7T^{2} \) |
| 11 | \( 1 - 3.26T + 11T^{2} \) |
| 13 | \( 1 - 2.63T + 13T^{2} \) |
| 19 | \( 1 - 3.97T + 19T^{2} \) |
| 23 | \( 1 - 2.46T + 23T^{2} \) |
| 29 | \( 1 - 1.70T + 29T^{2} \) |
| 31 | \( 1 + 2T + 31T^{2} \) |
| 37 | \( 1 + 10.2T + 37T^{2} \) |
| 41 | \( 1 - 6.70T + 41T^{2} \) |
| 43 | \( 1 - 0.921T + 43T^{2} \) |
| 47 | \( 1 + 5.70T + 47T^{2} \) |
| 53 | \( 1 - 8.03T + 53T^{2} \) |
| 61 | \( 1 + 4.73T + 61T^{2} \) |
| 67 | \( 1 + 9.70T + 67T^{2} \) |
| 71 | \( 1 - 12.2T + 71T^{2} \) |
| 73 | \( 1 - 3.29T + 73T^{2} \) |
| 79 | \( 1 + 13.8T + 79T^{2} \) |
| 83 | \( 1 + 12.8T + 83T^{2} \) |
| 89 | \( 1 + 10.2T + 89T^{2} \) |
| 97 | \( 1 + 9T + 97T^{2} \) |
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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
−8.340365119259009982276595891550, −7.41609238469507455152887884809, −6.87304385681632248666548337548, −5.96711737475779556465354158217, −5.34635942423830407206664625857, −4.45466054222496036127383963286, −3.50075244519821392922104088434, −2.48142612283272561415117130439, −1.58273255296681098936169004989, −1.26123698876824587865549024318,
1.26123698876824587865549024318, 1.58273255296681098936169004989, 2.48142612283272561415117130439, 3.50075244519821392922104088434, 4.45466054222496036127383963286, 5.34635942423830407206664625857, 5.96711737475779556465354158217, 6.87304385681632248666548337548, 7.41609238469507455152887884809, 8.340365119259009982276595891550