| L(s) = 1 | − 0.749·2-s + 3-s − 1.43·4-s − 5-s − 0.749·6-s − 2.59·7-s + 2.57·8-s + 9-s + 0.749·10-s − 4.18·11-s − 1.43·12-s − 4.28·13-s + 1.94·14-s − 15-s + 0.946·16-s − 7.23·17-s − 0.749·18-s − 0.407·19-s + 1.43·20-s − 2.59·21-s + 3.13·22-s − 4.76·23-s + 2.57·24-s + 25-s + 3.21·26-s + 27-s + 3.73·28-s + ⋯ |
| L(s) = 1 | − 0.529·2-s + 0.577·3-s − 0.719·4-s − 0.447·5-s − 0.305·6-s − 0.980·7-s + 0.910·8-s + 0.333·9-s + 0.236·10-s − 1.26·11-s − 0.415·12-s − 1.18·13-s + 0.519·14-s − 0.258·15-s + 0.236·16-s − 1.75·17-s − 0.176·18-s − 0.0934·19-s + 0.321·20-s − 0.565·21-s + 0.668·22-s − 0.992·23-s + 0.525·24-s + 0.200·25-s + 0.629·26-s + 0.192·27-s + 0.704·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.07760492553\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.07760492553\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 401 | \( 1 + T \) |
| good | 2 | \( 1 + 0.749T + 2T^{2} \) |
| 7 | \( 1 + 2.59T + 7T^{2} \) |
| 11 | \( 1 + 4.18T + 11T^{2} \) |
| 13 | \( 1 + 4.28T + 13T^{2} \) |
| 17 | \( 1 + 7.23T + 17T^{2} \) |
| 19 | \( 1 + 0.407T + 19T^{2} \) |
| 23 | \( 1 + 4.76T + 23T^{2} \) |
| 29 | \( 1 + 10.0T + 29T^{2} \) |
| 31 | \( 1 + 1.40T + 31T^{2} \) |
| 37 | \( 1 + 5.24T + 37T^{2} \) |
| 41 | \( 1 - 5.73T + 41T^{2} \) |
| 43 | \( 1 - 7.87T + 43T^{2} \) |
| 47 | \( 1 + 0.842T + 47T^{2} \) |
| 53 | \( 1 - 1.98T + 53T^{2} \) |
| 59 | \( 1 + 6.07T + 59T^{2} \) |
| 61 | \( 1 + 10.2T + 61T^{2} \) |
| 67 | \( 1 + 5.34T + 67T^{2} \) |
| 71 | \( 1 + 2.25T + 71T^{2} \) |
| 73 | \( 1 - 14.9T + 73T^{2} \) |
| 79 | \( 1 + 11.7T + 79T^{2} \) |
| 83 | \( 1 - 8.29T + 83T^{2} \) |
| 89 | \( 1 - 7.34T + 89T^{2} \) |
| 97 | \( 1 + 0.996T + 97T^{2} \) |
| 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
−7.991379686073248498647068246647, −7.59950165190261060296362875164, −6.99234643982512843660129602249, −5.96030490602159562206597719125, −5.06201081871559677395119123038, −4.35292849527075003605748015235, −3.69201122711655563354570675070, −2.68845084039963354866492836728, −1.96883908016850808180291795555, −0.14396322069297648901542733680,
0.14396322069297648901542733680, 1.96883908016850808180291795555, 2.68845084039963354866492836728, 3.69201122711655563354570675070, 4.35292849527075003605748015235, 5.06201081871559677395119123038, 5.96030490602159562206597719125, 6.99234643982512843660129602249, 7.59950165190261060296362875164, 7.991379686073248498647068246647
