| L(s) = 1 | − 2-s + 4-s − 2.95·5-s + 2.46·7-s − 8-s + 2.95·10-s + 0.814·11-s + 5.11·13-s − 2.46·14-s + 16-s − 3.17·17-s − 4.28·19-s − 2.95·20-s − 0.814·22-s − 3.02·23-s + 3.73·25-s − 5.11·26-s + 2.46·28-s − 0.136·29-s + 4.98·31-s − 32-s + 3.17·34-s − 7.28·35-s + 8.76·37-s + 4.28·38-s + 2.95·40-s + 10.5·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s − 1.32·5-s + 0.932·7-s − 0.353·8-s + 0.934·10-s + 0.245·11-s + 1.41·13-s − 0.659·14-s + 0.250·16-s − 0.769·17-s − 0.982·19-s − 0.660·20-s − 0.173·22-s − 0.631·23-s + 0.746·25-s − 1.00·26-s + 0.466·28-s − 0.0253·29-s + 0.896·31-s − 0.176·32-s + 0.543·34-s − 1.23·35-s + 1.44·37-s + 0.694·38-s + 0.467·40-s + 1.64·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8046 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8046 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.125848932\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.125848932\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 149 | \( 1 + T \) |
| good | 5 | \( 1 + 2.95T + 5T^{2} \) |
| 7 | \( 1 - 2.46T + 7T^{2} \) |
| 11 | \( 1 - 0.814T + 11T^{2} \) |
| 13 | \( 1 - 5.11T + 13T^{2} \) |
| 17 | \( 1 + 3.17T + 17T^{2} \) |
| 19 | \( 1 + 4.28T + 19T^{2} \) |
| 23 | \( 1 + 3.02T + 23T^{2} \) |
| 29 | \( 1 + 0.136T + 29T^{2} \) |
| 31 | \( 1 - 4.98T + 31T^{2} \) |
| 37 | \( 1 - 8.76T + 37T^{2} \) |
| 41 | \( 1 - 10.5T + 41T^{2} \) |
| 43 | \( 1 + 11.8T + 43T^{2} \) |
| 47 | \( 1 + 6.64T + 47T^{2} \) |
| 53 | \( 1 - 10.5T + 53T^{2} \) |
| 59 | \( 1 + 1.90T + 59T^{2} \) |
| 61 | \( 1 - 13.2T + 61T^{2} \) |
| 67 | \( 1 + 7.86T + 67T^{2} \) |
| 71 | \( 1 + 5.04T + 71T^{2} \) |
| 73 | \( 1 + 2.80T + 73T^{2} \) |
| 79 | \( 1 - 12.5T + 79T^{2} \) |
| 83 | \( 1 + 11.6T + 83T^{2} \) |
| 89 | \( 1 - 10.7T + 89T^{2} \) |
| 97 | \( 1 + 4.75T + 97T^{2} \) |
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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
−7.985626089109864065336650891869, −7.39592607260025825094030614314, −6.48459740826297314261642229818, −6.05775821035465508311570677447, −4.83787516302350721676527102411, −4.18091958214243645624082456537, −3.65845309439906986027495668484, −2.53130895616824386681405682004, −1.57510394837898366797643062879, −0.60854695244027595594082024578,
0.60854695244027595594082024578, 1.57510394837898366797643062879, 2.53130895616824386681405682004, 3.65845309439906986027495668484, 4.18091958214243645624082456537, 4.83787516302350721676527102411, 6.05775821035465508311570677447, 6.48459740826297314261642229818, 7.39592607260025825094030614314, 7.985626089109864065336650891869
