| L(s) = 1 | − 0.120·2-s − 2.76·3-s − 1.98·4-s + 2.80·5-s + 0.333·6-s + 0.479·8-s + 4.66·9-s − 0.338·10-s + 5.49·12-s + 1.07·13-s − 7.77·15-s + 3.91·16-s + 6.95·17-s − 0.560·18-s + 7.54·19-s − 5.57·20-s + 4.82·23-s − 1.32·24-s + 2.88·25-s − 0.129·26-s − 4.59·27-s − 1.22·29-s + 0.935·30-s + 8.07·31-s − 1.43·32-s − 0.837·34-s − 9.25·36-s + ⋯ |
| L(s) = 1 | − 0.0851·2-s − 1.59·3-s − 0.992·4-s + 1.25·5-s + 0.136·6-s + 0.169·8-s + 1.55·9-s − 0.106·10-s + 1.58·12-s + 0.299·13-s − 2.00·15-s + 0.978·16-s + 1.68·17-s − 0.132·18-s + 1.73·19-s − 1.24·20-s + 1.00·23-s − 0.271·24-s + 0.577·25-s − 0.0254·26-s − 0.884·27-s − 0.227·29-s + 0.170·30-s + 1.45·31-s − 0.252·32-s − 0.143·34-s − 1.54·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5929 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5929 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.452700985\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.452700985\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 0.120T + 2T^{2} \) |
| 3 | \( 1 + 2.76T + 3T^{2} \) |
| 5 | \( 1 - 2.80T + 5T^{2} \) |
| 13 | \( 1 - 1.07T + 13T^{2} \) |
| 17 | \( 1 - 6.95T + 17T^{2} \) |
| 19 | \( 1 - 7.54T + 19T^{2} \) |
| 23 | \( 1 - 4.82T + 23T^{2} \) |
| 29 | \( 1 + 1.22T + 29T^{2} \) |
| 31 | \( 1 - 8.07T + 31T^{2} \) |
| 37 | \( 1 - 1.53T + 37T^{2} \) |
| 41 | \( 1 - 9.29T + 41T^{2} \) |
| 43 | \( 1 - 5.23T + 43T^{2} \) |
| 47 | \( 1 - 1.89T + 47T^{2} \) |
| 53 | \( 1 + 3.82T + 53T^{2} \) |
| 59 | \( 1 - 6.66T + 59T^{2} \) |
| 61 | \( 1 + 9.79T + 61T^{2} \) |
| 67 | \( 1 + 2.06T + 67T^{2} \) |
| 71 | \( 1 - 12.5T + 71T^{2} \) |
| 73 | \( 1 - 2.56T + 73T^{2} \) |
| 79 | \( 1 - 15.3T + 79T^{2} \) |
| 83 | \( 1 - 2.04T + 83T^{2} \) |
| 89 | \( 1 + 4.76T + 89T^{2} \) |
| 97 | \( 1 + 9.11T + 97T^{2} \) |
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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
−7.952716421471303775880572574488, −7.35497545166321610662078306846, −6.30578517727784518619620723652, −5.84194030599593600015457006059, −5.24841094100958887335276885885, −4.92268930492129793503946555596, −3.82848754369734017313469954801, −2.82099703431472986318235076406, −1.18028935009977783833866920569, −0.922536110051565666393369596532,
0.922536110051565666393369596532, 1.18028935009977783833866920569, 2.82099703431472986318235076406, 3.82848754369734017313469954801, 4.92268930492129793503946555596, 5.24841094100958887335276885885, 5.84194030599593600015457006059, 6.30578517727784518619620723652, 7.35497545166321610662078306846, 7.952716421471303775880572574488
