L(s) = 1 | − 1.80·2-s − 3-s + 1.24·4-s − 2.07·5-s + 1.80·6-s − 7-s + 1.35·8-s + 9-s + 3.74·10-s + 5.11·11-s − 1.24·12-s + 1.80·14-s + 2.07·15-s − 4.93·16-s − 0.185·17-s − 1.80·18-s − 0.591·19-s − 2.59·20-s + 21-s − 9.21·22-s − 2.78·23-s − 1.35·24-s − 0.681·25-s − 27-s − 1.24·28-s − 9.95·29-s − 3.74·30-s + ⋯ |
L(s) = 1 | − 1.27·2-s − 0.577·3-s + 0.623·4-s − 0.929·5-s + 0.735·6-s − 0.377·7-s + 0.479·8-s + 0.333·9-s + 1.18·10-s + 1.54·11-s − 0.359·12-s + 0.481·14-s + 0.536·15-s − 1.23·16-s − 0.0448·17-s − 0.424·18-s − 0.135·19-s − 0.579·20-s + 0.218·21-s − 1.96·22-s − 0.581·23-s − 0.276·24-s − 0.136·25-s − 0.192·27-s − 0.235·28-s − 1.84·29-s − 0.683·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + 1.80T + 2T^{2} \) |
| 5 | \( 1 + 2.07T + 5T^{2} \) |
| 11 | \( 1 - 5.11T + 11T^{2} \) |
| 17 | \( 1 + 0.185T + 17T^{2} \) |
| 19 | \( 1 + 0.591T + 19T^{2} \) |
| 23 | \( 1 + 2.78T + 23T^{2} \) |
| 29 | \( 1 + 9.95T + 29T^{2} \) |
| 31 | \( 1 - 5.23T + 31T^{2} \) |
| 37 | \( 1 + 2.97T + 37T^{2} \) |
| 41 | \( 1 + 10.0T + 41T^{2} \) |
| 43 | \( 1 - 5.25T + 43T^{2} \) |
| 47 | \( 1 - 12.6T + 47T^{2} \) |
| 53 | \( 1 + 5.51T + 53T^{2} \) |
| 59 | \( 1 - 6.46T + 59T^{2} \) |
| 61 | \( 1 - 8.43T + 61T^{2} \) |
| 67 | \( 1 - 9.58T + 67T^{2} \) |
| 71 | \( 1 + 1.60T + 71T^{2} \) |
| 73 | \( 1 - 9.17T + 73T^{2} \) |
| 79 | \( 1 - 5.42T + 79T^{2} \) |
| 83 | \( 1 - 3.99T + 83T^{2} \) |
| 89 | \( 1 + 9.63T + 89T^{2} \) |
| 97 | \( 1 + 3.05T + 97T^{2} \) |
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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
−8.274879959625045190296326467226, −7.53035441419737237936538331277, −6.91497097391530183566570701999, −6.27397771905541787657360576386, −5.21253089611271693526778109275, −4.04904185215980668875521215313, −3.78059969023679556829698775415, −2.09107497882958714991519526211, −1.02553360461117411126271039035, 0,
1.02553360461117411126271039035, 2.09107497882958714991519526211, 3.78059969023679556829698775415, 4.04904185215980668875521215313, 5.21253089611271693526778109275, 6.27397771905541787657360576386, 6.91497097391530183566570701999, 7.53035441419737237936538331277, 8.274879959625045190296326467226