| L(s) = 1 | − 2-s − 0.127·3-s + 4-s − 4.01·5-s + 0.127·6-s + 0.758·7-s − 8-s − 2.98·9-s + 4.01·10-s + 11-s − 0.127·12-s + 1.31·13-s − 0.758·14-s + 0.510·15-s + 16-s + 6.43·17-s + 2.98·18-s − 2.17·19-s − 4.01·20-s − 0.0963·21-s − 22-s + 1.38·23-s + 0.127·24-s + 11.1·25-s − 1.31·26-s + 0.760·27-s + 0.758·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.0733·3-s + 0.5·4-s − 1.79·5-s + 0.0518·6-s + 0.286·7-s − 0.353·8-s − 0.994·9-s + 1.26·10-s + 0.301·11-s − 0.0366·12-s + 0.364·13-s − 0.202·14-s + 0.131·15-s + 0.250·16-s + 1.56·17-s + 0.703·18-s − 0.498·19-s − 0.897·20-s − 0.0210·21-s − 0.213·22-s + 0.288·23-s + 0.0259·24-s + 2.22·25-s − 0.257·26-s + 0.146·27-s + 0.143·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4334 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4334 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5651753831\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5651753831\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 197 | \( 1 + T \) |
| good | 3 | \( 1 + 0.127T + 3T^{2} \) |
| 5 | \( 1 + 4.01T + 5T^{2} \) |
| 7 | \( 1 - 0.758T + 7T^{2} \) |
| 13 | \( 1 - 1.31T + 13T^{2} \) |
| 17 | \( 1 - 6.43T + 17T^{2} \) |
| 19 | \( 1 + 2.17T + 19T^{2} \) |
| 23 | \( 1 - 1.38T + 23T^{2} \) |
| 29 | \( 1 + 9.97T + 29T^{2} \) |
| 31 | \( 1 - 1.03T + 31T^{2} \) |
| 37 | \( 1 + 8.26T + 37T^{2} \) |
| 41 | \( 1 - 2.08T + 41T^{2} \) |
| 43 | \( 1 + 3.66T + 43T^{2} \) |
| 47 | \( 1 + 3.69T + 47T^{2} \) |
| 53 | \( 1 - 0.619T + 53T^{2} \) |
| 59 | \( 1 + 10.6T + 59T^{2} \) |
| 61 | \( 1 + 11.9T + 61T^{2} \) |
| 67 | \( 1 - 1.30T + 67T^{2} \) |
| 71 | \( 1 + 14.7T + 71T^{2} \) |
| 73 | \( 1 - 13.9T + 73T^{2} \) |
| 79 | \( 1 - 13.9T + 79T^{2} \) |
| 83 | \( 1 - 0.522T + 83T^{2} \) |
| 89 | \( 1 + 1.38T + 89T^{2} \) |
| 97 | \( 1 - 16.2T + 97T^{2} \) |
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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
−8.267253584848020766669890942071, −7.79714426450114351686957045717, −7.24915047039161148991240707239, −6.32087172283405739274384090757, −5.46611799852016965702492307816, −4.58214477318462454675856362944, −3.44409916915895627768327641960, −3.26462672149136196733718500993, −1.69763374339927547771595195915, −0.47327256185411884235303701334,
0.47327256185411884235303701334, 1.69763374339927547771595195915, 3.26462672149136196733718500993, 3.44409916915895627768327641960, 4.58214477318462454675856362944, 5.46611799852016965702492307816, 6.32087172283405739274384090757, 7.24915047039161148991240707239, 7.79714426450114351686957045717, 8.267253584848020766669890942071
