L(s) = 1 | + 2.26·2-s + 3.14·4-s + 2.83·5-s − 4.72·7-s + 2.60·8-s + 6.43·10-s + 11-s − 0.329·13-s − 10.7·14-s − 0.388·16-s − 6.43·17-s − 6.41·19-s + 8.92·20-s + 2.26·22-s − 0.000241·23-s + 3.03·25-s − 0.747·26-s − 14.8·28-s − 8.79·29-s + 5.66·31-s − 6.08·32-s − 14.6·34-s − 13.4·35-s − 0.385·37-s − 14.5·38-s + 7.37·40-s − 4.58·41-s + ⋯ |
L(s) = 1 | + 1.60·2-s + 1.57·4-s + 1.26·5-s − 1.78·7-s + 0.920·8-s + 2.03·10-s + 0.301·11-s − 0.0913·13-s − 2.86·14-s − 0.0971·16-s − 1.56·17-s − 1.47·19-s + 1.99·20-s + 0.483·22-s − 5.04e − 5·23-s + 0.607·25-s − 0.146·26-s − 2.81·28-s − 1.63·29-s + 1.01·31-s − 1.07·32-s − 2.50·34-s − 2.26·35-s − 0.0633·37-s − 2.36·38-s + 1.16·40-s − 0.716·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 - T \) |
| 61 | \( 1 + T \) |
good | 2 | \( 1 - 2.26T + 2T^{2} \) |
| 5 | \( 1 - 2.83T + 5T^{2} \) |
| 7 | \( 1 + 4.72T + 7T^{2} \) |
| 13 | \( 1 + 0.329T + 13T^{2} \) |
| 17 | \( 1 + 6.43T + 17T^{2} \) |
| 19 | \( 1 + 6.41T + 19T^{2} \) |
| 23 | \( 1 + 0.000241T + 23T^{2} \) |
| 29 | \( 1 + 8.79T + 29T^{2} \) |
| 31 | \( 1 - 5.66T + 31T^{2} \) |
| 37 | \( 1 + 0.385T + 37T^{2} \) |
| 41 | \( 1 + 4.58T + 41T^{2} \) |
| 43 | \( 1 + 2.75T + 43T^{2} \) |
| 47 | \( 1 - 4.95T + 47T^{2} \) |
| 53 | \( 1 - 2.98T + 53T^{2} \) |
| 59 | \( 1 + 11.8T + 59T^{2} \) |
| 67 | \( 1 + 7.49T + 67T^{2} \) |
| 71 | \( 1 - 3.18T + 71T^{2} \) |
| 73 | \( 1 - 12.3T + 73T^{2} \) |
| 79 | \( 1 - 5.72T + 79T^{2} \) |
| 83 | \( 1 - 10.3T + 83T^{2} \) |
| 89 | \( 1 - 3.95T + 89T^{2} \) |
| 97 | \( 1 + 11.5T + 97T^{2} \) |
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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
−7.14118824355598133816180620233, −6.55279749931437910883911515338, −6.23412172965852055578348243860, −5.72784324494550788902917238370, −4.78644199354832402238942314038, −4.05572224993485090511795253874, −3.35295279090300127765568011835, −2.46306062118150653482242738237, −1.98844833206833517194483469456, 0,
1.98844833206833517194483469456, 2.46306062118150653482242738237, 3.35295279090300127765568011835, 4.05572224993485090511795253874, 4.78644199354832402238942314038, 5.72784324494550788902917238370, 6.23412172965852055578348243860, 6.55279749931437910883911515338, 7.14118824355598133816180620233