| L(s) = 1 | − 5-s + 2.33·7-s − 6.01·11-s + 2.44·13-s − 0.800·17-s + 0.800·19-s + 2.81·23-s + 25-s − 1.09·29-s − 7.48·31-s − 2.33·35-s − 37-s + 9.97·41-s + 2.90·43-s + 13.1·47-s − 1.54·49-s − 11.9·53-s + 6.01·55-s − 13.0·59-s − 5.21·61-s − 2.44·65-s − 3.47·67-s + 9.90·71-s + 0.948·73-s − 14.0·77-s − 5.40·79-s + 5.46·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 0.882·7-s − 1.81·11-s + 0.676·13-s − 0.194·17-s + 0.183·19-s + 0.587·23-s + 0.200·25-s − 0.203·29-s − 1.34·31-s − 0.394·35-s − 0.164·37-s + 1.55·41-s + 0.443·43-s + 1.91·47-s − 0.220·49-s − 1.63·53-s + 0.811·55-s − 1.70·59-s − 0.667·61-s − 0.302·65-s − 0.424·67-s + 1.17·71-s + 0.111·73-s − 1.60·77-s − 0.608·79-s + 0.599·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6660 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6660 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 37 | \( 1 + T \) |
| good | 7 | \( 1 - 2.33T + 7T^{2} \) |
| 11 | \( 1 + 6.01T + 11T^{2} \) |
| 13 | \( 1 - 2.44T + 13T^{2} \) |
| 17 | \( 1 + 0.800T + 17T^{2} \) |
| 19 | \( 1 - 0.800T + 19T^{2} \) |
| 23 | \( 1 - 2.81T + 23T^{2} \) |
| 29 | \( 1 + 1.09T + 29T^{2} \) |
| 31 | \( 1 + 7.48T + 31T^{2} \) |
| 41 | \( 1 - 9.97T + 41T^{2} \) |
| 43 | \( 1 - 2.90T + 43T^{2} \) |
| 47 | \( 1 - 13.1T + 47T^{2} \) |
| 53 | \( 1 + 11.9T + 53T^{2} \) |
| 59 | \( 1 + 13.0T + 59T^{2} \) |
| 61 | \( 1 + 5.21T + 61T^{2} \) |
| 67 | \( 1 + 3.47T + 67T^{2} \) |
| 71 | \( 1 - 9.90T + 71T^{2} \) |
| 73 | \( 1 - 0.948T + 73T^{2} \) |
| 79 | \( 1 + 5.40T + 79T^{2} \) |
| 83 | \( 1 - 5.46T + 83T^{2} \) |
| 89 | \( 1 + 9.59T + 89T^{2} \) |
| 97 | \( 1 + 8.41T + 97T^{2} \) |
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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
−7.68836711581303228983847006370, −7.23505023143023552156780677486, −6.09354665957523066613936613421, −5.43701475620680838839554039601, −4.82170835437489998225101257144, −4.07902091519910148099207061870, −3.11491108742672552727741721432, −2.35136708588951231259668504209, −1.29109764722590564755194409368, 0,
1.29109764722590564755194409368, 2.35136708588951231259668504209, 3.11491108742672552727741721432, 4.07902091519910148099207061870, 4.82170835437489998225101257144, 5.43701475620680838839554039601, 6.09354665957523066613936613421, 7.23505023143023552156780677486, 7.68836711581303228983847006370
