| L(s) = 1 | + 2-s + 2.39·3-s + 4-s − 3.86·5-s + 2.39·6-s − 3.67·7-s + 8-s + 2.72·9-s − 3.86·10-s + 2.25·11-s + 2.39·12-s − 5.43·13-s − 3.67·14-s − 9.25·15-s + 16-s + 6.26·17-s + 2.72·18-s + 2.81·19-s − 3.86·20-s − 8.78·21-s + 2.25·22-s + 2.64·23-s + 2.39·24-s + 9.95·25-s − 5.43·26-s − 0.649·27-s − 3.67·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.38·3-s + 0.5·4-s − 1.72·5-s + 0.977·6-s − 1.38·7-s + 0.353·8-s + 0.909·9-s − 1.22·10-s + 0.680·11-s + 0.690·12-s − 1.50·13-s − 0.981·14-s − 2.38·15-s + 0.250·16-s + 1.52·17-s + 0.643·18-s + 0.645·19-s − 0.864·20-s − 1.91·21-s + 0.480·22-s + 0.550·23-s + 0.488·24-s + 1.99·25-s − 1.06·26-s − 0.124·27-s − 0.693·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4022 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4022 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.059483517\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.059483517\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 2011 | \( 1 + T \) |
| good | 3 | \( 1 - 2.39T + 3T^{2} \) |
| 5 | \( 1 + 3.86T + 5T^{2} \) |
| 7 | \( 1 + 3.67T + 7T^{2} \) |
| 11 | \( 1 - 2.25T + 11T^{2} \) |
| 13 | \( 1 + 5.43T + 13T^{2} \) |
| 17 | \( 1 - 6.26T + 17T^{2} \) |
| 19 | \( 1 - 2.81T + 19T^{2} \) |
| 23 | \( 1 - 2.64T + 23T^{2} \) |
| 29 | \( 1 - 5.80T + 29T^{2} \) |
| 31 | \( 1 - 5.79T + 31T^{2} \) |
| 37 | \( 1 - 1.68T + 37T^{2} \) |
| 41 | \( 1 - 0.537T + 41T^{2} \) |
| 43 | \( 1 - 0.829T + 43T^{2} \) |
| 47 | \( 1 - 3.53T + 47T^{2} \) |
| 53 | \( 1 - 7.30T + 53T^{2} \) |
| 59 | \( 1 - 14.4T + 59T^{2} \) |
| 61 | \( 1 + 11.0T + 61T^{2} \) |
| 67 | \( 1 + 2.63T + 67T^{2} \) |
| 71 | \( 1 - 14.4T + 71T^{2} \) |
| 73 | \( 1 - 1.38T + 73T^{2} \) |
| 79 | \( 1 + 15.8T + 79T^{2} \) |
| 83 | \( 1 + 6.46T + 83T^{2} \) |
| 89 | \( 1 - 6.46T + 89T^{2} \) |
| 97 | \( 1 - 12.8T + 97T^{2} \) |
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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
−8.300552800356158985532802232472, −7.55849834632572187228556760886, −7.24497461785631016902765504661, −6.45756757885760381896977141412, −5.25320010683834405351442086312, −4.30936108168955315716260401794, −3.67938300098322335850729008043, −3.06691282218327792216263109103, −2.66513876275451714384348709849, −0.832291645651327212202050608036,
0.832291645651327212202050608036, 2.66513876275451714384348709849, 3.06691282218327792216263109103, 3.67938300098322335850729008043, 4.30936108168955315716260401794, 5.25320010683834405351442086312, 6.45756757885760381896977141412, 7.24497461785631016902765504661, 7.55849834632572187228556760886, 8.300552800356158985532802232472
