L(s) = 1 | − 2-s + 1.50·3-s + 4-s − 2.72·5-s − 1.50·6-s − 2.36·7-s − 8-s − 0.734·9-s + 2.72·10-s + 3.29·11-s + 1.50·12-s − 0.0258·13-s + 2.36·14-s − 4.10·15-s + 16-s + 1.25·17-s + 0.734·18-s − 1.42·19-s − 2.72·20-s − 3.55·21-s − 3.29·22-s + 23-s − 1.50·24-s + 2.42·25-s + 0.0258·26-s − 5.62·27-s − 2.36·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.868·3-s + 0.5·4-s − 1.21·5-s − 0.614·6-s − 0.893·7-s − 0.353·8-s − 0.244·9-s + 0.861·10-s + 0.992·11-s + 0.434·12-s − 0.00718·13-s + 0.632·14-s − 1.05·15-s + 0.250·16-s + 0.304·17-s + 0.173·18-s − 0.327·19-s − 0.609·20-s − 0.776·21-s − 0.702·22-s + 0.208·23-s − 0.307·24-s + 0.485·25-s + 0.00507·26-s − 1.08·27-s − 0.446·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| 131 | \( 1 - T \) |
good | 3 | \( 1 - 1.50T + 3T^{2} \) |
| 5 | \( 1 + 2.72T + 5T^{2} \) |
| 7 | \( 1 + 2.36T + 7T^{2} \) |
| 11 | \( 1 - 3.29T + 11T^{2} \) |
| 13 | \( 1 + 0.0258T + 13T^{2} \) |
| 17 | \( 1 - 1.25T + 17T^{2} \) |
| 19 | \( 1 + 1.42T + 19T^{2} \) |
| 29 | \( 1 - 1.18T + 29T^{2} \) |
| 31 | \( 1 - 5.97T + 31T^{2} \) |
| 37 | \( 1 - 6.53T + 37T^{2} \) |
| 41 | \( 1 - 1.11T + 41T^{2} \) |
| 43 | \( 1 - 3.03T + 43T^{2} \) |
| 47 | \( 1 - 11.9T + 47T^{2} \) |
| 53 | \( 1 + 8.79T + 53T^{2} \) |
| 59 | \( 1 + 0.637T + 59T^{2} \) |
| 61 | \( 1 + 10.4T + 61T^{2} \) |
| 67 | \( 1 - 4.76T + 67T^{2} \) |
| 71 | \( 1 + 7.69T + 71T^{2} \) |
| 73 | \( 1 + 6.63T + 73T^{2} \) |
| 79 | \( 1 - 7.08T + 79T^{2} \) |
| 83 | \( 1 + 13.3T + 83T^{2} \) |
| 89 | \( 1 - 11.0T + 89T^{2} \) |
| 97 | \( 1 - 2.00T + 97T^{2} \) |
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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
−7.82415378257529382564008149830, −7.31176114789711373815654908554, −6.45302401976823426956736463605, −5.91011355664431032733415258794, −4.50585307974997526877587472474, −3.82500040625599956058597140588, −3.14931789792027318760802562324, −2.50454967292491822434559229863, −1.14982863448888390230547221877, 0,
1.14982863448888390230547221877, 2.50454967292491822434559229863, 3.14931789792027318760802562324, 3.82500040625599956058597140588, 4.50585307974997526877587472474, 5.91011355664431032733415258794, 6.45302401976823426956736463605, 7.31176114789711373815654908554, 7.82415378257529382564008149830