| L(s) = 1 | + 2-s + 0.717·3-s + 4-s + 2.91·5-s + 0.717·6-s − 7-s + 8-s − 2.48·9-s + 2.91·10-s + 2.56·11-s + 0.717·12-s + 1.75·13-s − 14-s + 2.09·15-s + 16-s + 1.39·17-s − 2.48·18-s + 2.91·20-s − 0.717·21-s + 2.56·22-s − 3.44·23-s + 0.717·24-s + 3.50·25-s + 1.75·26-s − 3.93·27-s − 28-s − 0.789·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.414·3-s + 0.5·4-s + 1.30·5-s + 0.293·6-s − 0.377·7-s + 0.353·8-s − 0.828·9-s + 0.922·10-s + 0.773·11-s + 0.207·12-s + 0.487·13-s − 0.267·14-s + 0.540·15-s + 0.250·16-s + 0.338·17-s − 0.585·18-s + 0.652·20-s − 0.156·21-s + 0.546·22-s − 0.717·23-s + 0.146·24-s + 0.700·25-s + 0.344·26-s − 0.757·27-s − 0.188·28-s − 0.146·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5054 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5054 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.705914240\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.705914240\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 - 0.717T + 3T^{2} \) |
| 5 | \( 1 - 2.91T + 5T^{2} \) |
| 11 | \( 1 - 2.56T + 11T^{2} \) |
| 13 | \( 1 - 1.75T + 13T^{2} \) |
| 17 | \( 1 - 1.39T + 17T^{2} \) |
| 23 | \( 1 + 3.44T + 23T^{2} \) |
| 29 | \( 1 + 0.789T + 29T^{2} \) |
| 31 | \( 1 - 4.99T + 31T^{2} \) |
| 37 | \( 1 - 7.90T + 37T^{2} \) |
| 41 | \( 1 - 6.60T + 41T^{2} \) |
| 43 | \( 1 - 12.2T + 43T^{2} \) |
| 47 | \( 1 + 1.23T + 47T^{2} \) |
| 53 | \( 1 + 7.30T + 53T^{2} \) |
| 59 | \( 1 - 1.68T + 59T^{2} \) |
| 61 | \( 1 - 4.80T + 61T^{2} \) |
| 67 | \( 1 - 2.63T + 67T^{2} \) |
| 71 | \( 1 - 8.70T + 71T^{2} \) |
| 73 | \( 1 + 4.32T + 73T^{2} \) |
| 79 | \( 1 + 4.23T + 79T^{2} \) |
| 83 | \( 1 + 13.6T + 83T^{2} \) |
| 89 | \( 1 - 1.50T + 89T^{2} \) |
| 97 | \( 1 + 11.0T + 97T^{2} \) |
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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
−8.252312830934213990180463800310, −7.45247272342261026968465192665, −6.41934939743372502235422369477, −6.01095339813978223703056019558, −5.58807778914962536394148941589, −4.46617832558645120543556598916, −3.69837781798620995027316746914, −2.78750088108888910513630055719, −2.20537089232434850677212624316, −1.10186350511447918272927589107,
1.10186350511447918272927589107, 2.20537089232434850677212624316, 2.78750088108888910513630055719, 3.69837781798620995027316746914, 4.46617832558645120543556598916, 5.58807778914962536394148941589, 6.01095339813978223703056019558, 6.41934939743372502235422369477, 7.45247272342261026968465192665, 8.252312830934213990180463800310
