L(s) = 1 | + 1.66·2-s + 0.787·4-s − 3.19·5-s − 2.48·7-s − 2.02·8-s − 5.33·10-s + 4.44·11-s − 0.0955·13-s − 4.15·14-s − 4.95·16-s − 0.658·17-s + 5.40·19-s − 2.51·20-s + 7.42·22-s − 0.635·23-s + 5.21·25-s − 0.159·26-s − 1.95·28-s + 9.38·29-s + 7.21·31-s − 4.22·32-s − 1.09·34-s + 7.94·35-s − 11.9·37-s + 9.03·38-s + 6.46·40-s + 6.90·41-s + ⋯ |
L(s) = 1 | + 1.18·2-s + 0.393·4-s − 1.42·5-s − 0.939·7-s − 0.715·8-s − 1.68·10-s + 1.34·11-s − 0.0265·13-s − 1.10·14-s − 1.23·16-s − 0.159·17-s + 1.24·19-s − 0.562·20-s + 1.58·22-s − 0.132·23-s + 1.04·25-s − 0.0312·26-s − 0.369·28-s + 1.74·29-s + 1.29·31-s − 0.746·32-s − 0.188·34-s + 1.34·35-s − 1.96·37-s + 1.46·38-s + 1.02·40-s + 1.07·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.981952212\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.981952212\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 239 | \( 1 - T \) |
good | 2 | \( 1 - 1.66T + 2T^{2} \) |
| 5 | \( 1 + 3.19T + 5T^{2} \) |
| 7 | \( 1 + 2.48T + 7T^{2} \) |
| 11 | \( 1 - 4.44T + 11T^{2} \) |
| 13 | \( 1 + 0.0955T + 13T^{2} \) |
| 17 | \( 1 + 0.658T + 17T^{2} \) |
| 19 | \( 1 - 5.40T + 19T^{2} \) |
| 23 | \( 1 + 0.635T + 23T^{2} \) |
| 29 | \( 1 - 9.38T + 29T^{2} \) |
| 31 | \( 1 - 7.21T + 31T^{2} \) |
| 37 | \( 1 + 11.9T + 37T^{2} \) |
| 41 | \( 1 - 6.90T + 41T^{2} \) |
| 43 | \( 1 - 7.01T + 43T^{2} \) |
| 47 | \( 1 - 7.53T + 47T^{2} \) |
| 53 | \( 1 - 0.341T + 53T^{2} \) |
| 59 | \( 1 - 1.57T + 59T^{2} \) |
| 61 | \( 1 + 1.38T + 61T^{2} \) |
| 67 | \( 1 + 0.142T + 67T^{2} \) |
| 71 | \( 1 + 3.28T + 71T^{2} \) |
| 73 | \( 1 - 0.402T + 73T^{2} \) |
| 79 | \( 1 - 6.98T + 79T^{2} \) |
| 83 | \( 1 - 0.0483T + 83T^{2} \) |
| 89 | \( 1 - 5.95T + 89T^{2} \) |
| 97 | \( 1 + 11.1T + 97T^{2} \) |
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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
−9.036804982968724156901241032881, −8.315434053251254102958504684571, −7.24105650451266888415440146461, −6.64703265210471435109470951560, −5.90780785870434689910082752870, −4.79813667168021057322396564825, −4.11974959149869713775602498117, −3.50411803133667092308155800119, −2.82588728046883639046632303080, −0.77244657914673040894494676809,
0.77244657914673040894494676809, 2.82588728046883639046632303080, 3.50411803133667092308155800119, 4.11974959149869713775602498117, 4.79813667168021057322396564825, 5.90780785870434689910082752870, 6.64703265210471435109470951560, 7.24105650451266888415440146461, 8.315434053251254102958504684571, 9.036804982968724156901241032881