| L(s) = 1 | − 5-s + 1.89·7-s + 2.38·11-s − 4.38·13-s − 0.761·17-s + 0.864·19-s + 23-s + 25-s − 8.76·29-s + 9.35·31-s − 1.89·35-s − 0.103·37-s − 9.53·41-s + 3.25·43-s − 3.13·47-s − 3.40·49-s − 8.01·53-s − 2.38·55-s − 10.2·59-s + 3.64·61-s + 4.38·65-s + 12.6·67-s + 6.55·71-s − 4.92·73-s + 4.52·77-s − 7.04·79-s − 5.23·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 0.716·7-s + 0.719·11-s − 1.21·13-s − 0.184·17-s + 0.198·19-s + 0.208·23-s + 0.200·25-s − 1.62·29-s + 1.68·31-s − 0.320·35-s − 0.0169·37-s − 1.48·41-s + 0.495·43-s − 0.457·47-s − 0.485·49-s − 1.10·53-s − 0.321·55-s − 1.33·59-s + 0.466·61-s + 0.544·65-s + 1.54·67-s + 0.777·71-s − 0.576·73-s + 0.516·77-s − 0.792·79-s − 0.574·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| good | 7 | \( 1 - 1.89T + 7T^{2} \) |
| 11 | \( 1 - 2.38T + 11T^{2} \) |
| 13 | \( 1 + 4.38T + 13T^{2} \) |
| 17 | \( 1 + 0.761T + 17T^{2} \) |
| 19 | \( 1 - 0.864T + 19T^{2} \) |
| 29 | \( 1 + 8.76T + 29T^{2} \) |
| 31 | \( 1 - 9.35T + 31T^{2} \) |
| 37 | \( 1 + 0.103T + 37T^{2} \) |
| 41 | \( 1 + 9.53T + 41T^{2} \) |
| 43 | \( 1 - 3.25T + 43T^{2} \) |
| 47 | \( 1 + 3.13T + 47T^{2} \) |
| 53 | \( 1 + 8.01T + 53T^{2} \) |
| 59 | \( 1 + 10.2T + 59T^{2} \) |
| 61 | \( 1 - 3.64T + 61T^{2} \) |
| 67 | \( 1 - 12.6T + 67T^{2} \) |
| 71 | \( 1 - 6.55T + 71T^{2} \) |
| 73 | \( 1 + 4.92T + 73T^{2} \) |
| 79 | \( 1 + 7.04T + 79T^{2} \) |
| 83 | \( 1 + 5.23T + 83T^{2} \) |
| 89 | \( 1 - 10.9T + 89T^{2} \) |
| 97 | \( 1 + 0.0645T + 97T^{2} \) |
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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
−7.52158682001763146851069361883, −6.85288096661261641314842598299, −6.17672378124636707675330975970, −5.13196402492374083254219720806, −4.75447834965650440564772551469, −3.94207286375282410138204053169, −3.12516962424246226111322153637, −2.16934184795864000058414652819, −1.29436438289352450093246669339, 0,
1.29436438289352450093246669339, 2.16934184795864000058414652819, 3.12516962424246226111322153637, 3.94207286375282410138204053169, 4.75447834965650440564772551469, 5.13196402492374083254219720806, 6.17672378124636707675330975970, 6.85288096661261641314842598299, 7.52158682001763146851069361883
