| L(s) = 1 | − 3.40·7-s + 4.40·11-s + 2.06·13-s + 1.40·17-s − 6.35·19-s − 0.107·23-s − 9.08·29-s + 3.06·31-s + 1.95·37-s + 8.69·41-s − 7.12·43-s + 7.48·47-s + 4.57·49-s − 13.0·53-s + 13.1·59-s − 1.72·61-s − 12.3·67-s + 7.50·71-s − 5.42·73-s − 14.9·77-s + 9.43·79-s − 8.97·83-s − 4.01·89-s − 7.01·91-s + 2.17·97-s − 2.97·101-s + 6.63·103-s + ⋯ |
| L(s) = 1 | − 1.28·7-s + 1.32·11-s + 0.572·13-s + 0.339·17-s − 1.45·19-s − 0.0224·23-s − 1.68·29-s + 0.550·31-s + 0.321·37-s + 1.35·41-s − 1.08·43-s + 1.09·47-s + 0.653·49-s − 1.79·53-s + 1.71·59-s − 0.220·61-s − 1.51·67-s + 0.891·71-s − 0.634·73-s − 1.70·77-s + 1.06·79-s − 0.985·83-s − 0.425·89-s − 0.735·91-s + 0.220·97-s − 0.295·101-s + 0.653·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| good | 7 | \( 1 + 3.40T + 7T^{2} \) |
| 11 | \( 1 - 4.40T + 11T^{2} \) |
| 13 | \( 1 - 2.06T + 13T^{2} \) |
| 17 | \( 1 - 1.40T + 17T^{2} \) |
| 19 | \( 1 + 6.35T + 19T^{2} \) |
| 23 | \( 1 + 0.107T + 23T^{2} \) |
| 29 | \( 1 + 9.08T + 29T^{2} \) |
| 31 | \( 1 - 3.06T + 31T^{2} \) |
| 37 | \( 1 - 1.95T + 37T^{2} \) |
| 41 | \( 1 - 8.69T + 41T^{2} \) |
| 43 | \( 1 + 7.12T + 43T^{2} \) |
| 47 | \( 1 - 7.48T + 47T^{2} \) |
| 53 | \( 1 + 13.0T + 53T^{2} \) |
| 59 | \( 1 - 13.1T + 59T^{2} \) |
| 61 | \( 1 + 1.72T + 61T^{2} \) |
| 67 | \( 1 + 12.3T + 67T^{2} \) |
| 71 | \( 1 - 7.50T + 71T^{2} \) |
| 73 | \( 1 + 5.42T + 73T^{2} \) |
| 79 | \( 1 - 9.43T + 79T^{2} \) |
| 83 | \( 1 + 8.97T + 83T^{2} \) |
| 89 | \( 1 + 4.01T + 89T^{2} \) |
| 97 | \( 1 - 2.17T + 97T^{2} \) |
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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
−7.35034624730196846571017261232, −6.64056190703381891014953625297, −6.20831681884758661326595409888, −5.64834149698759137638269209933, −4.41853573805097903892239266886, −3.86631087657258397063155654324, −3.24832872956972015290753434734, −2.24089355602346076763201143035, −1.22383645751313171064137343074, 0,
1.22383645751313171064137343074, 2.24089355602346076763201143035, 3.24832872956972015290753434734, 3.86631087657258397063155654324, 4.41853573805097903892239266886, 5.64834149698759137638269209933, 6.20831681884758661326595409888, 6.64056190703381891014953625297, 7.35034624730196846571017261232
