| L(s) = 1 | + 3.10·3-s + 0.407·5-s + 3.05·7-s + 6.67·9-s + 2.16·11-s + 0.550·13-s + 1.26·15-s − 3.34·17-s − 6.78·19-s + 9.49·21-s + 5.22·23-s − 4.83·25-s + 11.4·27-s − 3.25·29-s + 8.03·31-s + 6.73·33-s + 1.24·35-s + 9.82·37-s + 1.71·39-s + 8.04·41-s − 3.91·43-s + 2.71·45-s − 47-s + 2.31·49-s − 10.3·51-s − 6.41·53-s + 0.882·55-s + ⋯ |
| L(s) = 1 | + 1.79·3-s + 0.182·5-s + 1.15·7-s + 2.22·9-s + 0.653·11-s + 0.152·13-s + 0.327·15-s − 0.810·17-s − 1.55·19-s + 2.07·21-s + 1.09·23-s − 0.966·25-s + 2.19·27-s − 0.603·29-s + 1.44·31-s + 1.17·33-s + 0.210·35-s + 1.61·37-s + 0.274·39-s + 1.25·41-s − 0.596·43-s + 0.405·45-s − 0.145·47-s + 0.331·49-s − 1.45·51-s − 0.881·53-s + 0.118·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.142676115\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.142676115\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 47 | \( 1 + T \) |
| good | 3 | \( 1 - 3.10T + 3T^{2} \) |
| 5 | \( 1 - 0.407T + 5T^{2} \) |
| 7 | \( 1 - 3.05T + 7T^{2} \) |
| 11 | \( 1 - 2.16T + 11T^{2} \) |
| 13 | \( 1 - 0.550T + 13T^{2} \) |
| 17 | \( 1 + 3.34T + 17T^{2} \) |
| 19 | \( 1 + 6.78T + 19T^{2} \) |
| 23 | \( 1 - 5.22T + 23T^{2} \) |
| 29 | \( 1 + 3.25T + 29T^{2} \) |
| 31 | \( 1 - 8.03T + 31T^{2} \) |
| 37 | \( 1 - 9.82T + 37T^{2} \) |
| 41 | \( 1 - 8.04T + 41T^{2} \) |
| 43 | \( 1 + 3.91T + 43T^{2} \) |
| 53 | \( 1 + 6.41T + 53T^{2} \) |
| 59 | \( 1 + 2.73T + 59T^{2} \) |
| 61 | \( 1 + 4.63T + 61T^{2} \) |
| 67 | \( 1 - 11.2T + 67T^{2} \) |
| 71 | \( 1 - 5.75T + 71T^{2} \) |
| 73 | \( 1 + 6.94T + 73T^{2} \) |
| 79 | \( 1 - 15.0T + 79T^{2} \) |
| 83 | \( 1 - 9.73T + 83T^{2} \) |
| 89 | \( 1 - 9.09T + 89T^{2} \) |
| 97 | \( 1 + 15.1T + 97T^{2} \) |
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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
−8.092584252992193782972303291807, −7.75172257779833422373296096966, −6.76780736079030256843821139540, −6.17821214004036167252495101583, −4.80833128939801294219771431400, −4.33653817474093762272179116607, −3.66291887984871986100060353699, −2.57424623452949383704523027198, −2.07533618118552063030324196353, −1.21366663447605279248425696340,
1.21366663447605279248425696340, 2.07533618118552063030324196353, 2.57424623452949383704523027198, 3.66291887984871986100060353699, 4.33653817474093762272179116607, 4.80833128939801294219771431400, 6.17821214004036167252495101583, 6.76780736079030256843821139540, 7.75172257779833422373296096966, 8.092584252992193782972303291807
