| L(s) = 1 | − 0.854·2-s − 1.26·4-s + 2.82·5-s + 3.33·7-s + 2.79·8-s − 2.41·10-s + 0.210·11-s − 4.81·13-s − 2.84·14-s + 0.150·16-s − 4.49·17-s − 4.63·19-s − 3.58·20-s − 0.179·22-s − 7.53·23-s + 2.95·25-s + 4.11·26-s − 4.22·28-s − 2.29·29-s − 7.14·31-s − 5.71·32-s + 3.83·34-s + 9.39·35-s − 1.94·37-s + 3.96·38-s + 7.88·40-s − 8.56·41-s + ⋯ |
| L(s) = 1 | − 0.604·2-s − 0.634·4-s + 1.26·5-s + 1.25·7-s + 0.987·8-s − 0.762·10-s + 0.0634·11-s − 1.33·13-s − 0.760·14-s + 0.0376·16-s − 1.08·17-s − 1.06·19-s − 0.800·20-s − 0.0383·22-s − 1.57·23-s + 0.590·25-s + 0.807·26-s − 0.799·28-s − 0.425·29-s − 1.28·31-s − 1.01·32-s + 0.658·34-s + 1.58·35-s − 0.320·37-s + 0.642·38-s + 1.24·40-s − 1.33·41-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
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 239 | \( 1 + T \) |
| good | 2 | \( 1 + 0.854T + 2T^{2} \) |
| 5 | \( 1 - 2.82T + 5T^{2} \) |
| 7 | \( 1 - 3.33T + 7T^{2} \) |
| 11 | \( 1 - 0.210T + 11T^{2} \) |
| 13 | \( 1 + 4.81T + 13T^{2} \) |
| 17 | \( 1 + 4.49T + 17T^{2} \) |
| 19 | \( 1 + 4.63T + 19T^{2} \) |
| 23 | \( 1 + 7.53T + 23T^{2} \) |
| 29 | \( 1 + 2.29T + 29T^{2} \) |
| 31 | \( 1 + 7.14T + 31T^{2} \) |
| 37 | \( 1 + 1.94T + 37T^{2} \) |
| 41 | \( 1 + 8.56T + 41T^{2} \) |
| 43 | \( 1 + 1.06T + 43T^{2} \) |
| 47 | \( 1 + 4.78T + 47T^{2} \) |
| 53 | \( 1 + 4.11T + 53T^{2} \) |
| 59 | \( 1 + 7.70T + 59T^{2} \) |
| 61 | \( 1 - 8.53T + 61T^{2} \) |
| 67 | \( 1 - 16.0T + 67T^{2} \) |
| 71 | \( 1 - 5.59T + 71T^{2} \) |
| 73 | \( 1 - 11.2T + 73T^{2} \) |
| 79 | \( 1 + 2.49T + 79T^{2} \) |
| 83 | \( 1 + 10.5T + 83T^{2} \) |
| 89 | \( 1 + 4.26T + 89T^{2} \) |
| 97 | \( 1 - 1.01T + 97T^{2} \) |
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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
−8.664212883178060916561153919961, −8.172911836129150831156603771631, −7.30125079740309602653101806087, −6.36911817869845580652154337875, −5.30416378973764711747671953197, −4.84375272352294101252926796877, −3.94419495073796829462172498568, −2.03909662811114627247668496976, −1.88378557177365249338484410473, 0,
1.88378557177365249338484410473, 2.03909662811114627247668496976, 3.94419495073796829462172498568, 4.84375272352294101252926796877, 5.30416378973764711747671953197, 6.36911817869845580652154337875, 7.30125079740309602653101806087, 8.172911836129150831156603771631, 8.664212883178060916561153919961
