| L(s) = 1 | + 0.231·2-s − 3-s − 1.94·4-s + 3.70·5-s − 0.231·6-s − 0.911·7-s − 0.912·8-s + 9-s + 0.857·10-s − 11-s + 1.94·12-s − 1.28·13-s − 0.210·14-s − 3.70·15-s + 3.68·16-s − 7.36·17-s + 0.231·18-s + 5.91·19-s − 7.21·20-s + 0.911·21-s − 0.231·22-s + 6.40·23-s + 0.912·24-s + 8.74·25-s − 0.298·26-s − 27-s + 1.77·28-s + ⋯ |
| L(s) = 1 | + 0.163·2-s − 0.577·3-s − 0.973·4-s + 1.65·5-s − 0.0944·6-s − 0.344·7-s − 0.322·8-s + 0.333·9-s + 0.271·10-s − 0.301·11-s + 0.561·12-s − 0.357·13-s − 0.0563·14-s − 0.957·15-s + 0.920·16-s − 1.78·17-s + 0.0545·18-s + 1.35·19-s − 1.61·20-s + 0.198·21-s − 0.0493·22-s + 1.33·23-s + 0.186·24-s + 1.74·25-s − 0.0585·26-s − 0.192·27-s + 0.335·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.503590449\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.503590449\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 61 | \( 1 - T \) |
| good | 2 | \( 1 - 0.231T + 2T^{2} \) |
| 5 | \( 1 - 3.70T + 5T^{2} \) |
| 7 | \( 1 + 0.911T + 7T^{2} \) |
| 13 | \( 1 + 1.28T + 13T^{2} \) |
| 17 | \( 1 + 7.36T + 17T^{2} \) |
| 19 | \( 1 - 5.91T + 19T^{2} \) |
| 23 | \( 1 - 6.40T + 23T^{2} \) |
| 29 | \( 1 - 9.58T + 29T^{2} \) |
| 31 | \( 1 + 8.65T + 31T^{2} \) |
| 37 | \( 1 + 8.74T + 37T^{2} \) |
| 41 | \( 1 - 6.54T + 41T^{2} \) |
| 43 | \( 1 - 9.59T + 43T^{2} \) |
| 47 | \( 1 - 5.82T + 47T^{2} \) |
| 53 | \( 1 + 5.01T + 53T^{2} \) |
| 59 | \( 1 - 1.55T + 59T^{2} \) |
| 67 | \( 1 - 12.7T + 67T^{2} \) |
| 71 | \( 1 + 0.506T + 71T^{2} \) |
| 73 | \( 1 + 3.71T + 73T^{2} \) |
| 79 | \( 1 + 5.28T + 79T^{2} \) |
| 83 | \( 1 - 7.43T + 83T^{2} \) |
| 89 | \( 1 - 12.7T + 89T^{2} \) |
| 97 | \( 1 - 19.0T + 97T^{2} \) |
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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
−9.153085994127430960419332570636, −8.829539470176102114525562864204, −7.39374499935565149168528110667, −6.59942252824593166421651146680, −5.82839002951911048151273430931, −5.13723391983602330876233826246, −4.62895320405394070437823711815, −3.25000757198434333766824160036, −2.19488637014103533077623999827, −0.836187038307320560632872665401,
0.836187038307320560632872665401, 2.19488637014103533077623999827, 3.25000757198434333766824160036, 4.62895320405394070437823711815, 5.13723391983602330876233826246, 5.82839002951911048151273430931, 6.59942252824593166421651146680, 7.39374499935565149168528110667, 8.829539470176102114525562864204, 9.153085994127430960419332570636
