| L(s) = 1 | − 2.11·2-s + 3-s + 2.47·4-s − 0.357·5-s − 2.11·6-s + 1.64·7-s − 1.00·8-s + 9-s + 0.756·10-s + 2.94·11-s + 2.47·12-s + 4.94·13-s − 3.47·14-s − 0.357·15-s − 2.83·16-s − 1.28·17-s − 2.11·18-s − 6.81·19-s − 0.885·20-s + 1.64·21-s − 6.22·22-s + 2.22·23-s − 1.00·24-s − 4.87·25-s − 10.4·26-s + 27-s + 4.06·28-s + ⋯ |
| L(s) = 1 | − 1.49·2-s + 0.577·3-s + 1.23·4-s − 0.160·5-s − 0.863·6-s + 0.620·7-s − 0.353·8-s + 0.333·9-s + 0.239·10-s + 0.888·11-s + 0.713·12-s + 1.37·13-s − 0.928·14-s − 0.0924·15-s − 0.707·16-s − 0.311·17-s − 0.498·18-s − 1.56·19-s − 0.197·20-s + 0.358·21-s − 1.32·22-s + 0.464·23-s − 0.204·24-s − 0.974·25-s − 2.05·26-s + 0.192·27-s + 0.767·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 93 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 93 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6389748509\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6389748509\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 31 | \( 1 + T \) |
| good | 2 | \( 1 + 2.11T + 2T^{2} \) |
| 5 | \( 1 + 0.357T + 5T^{2} \) |
| 7 | \( 1 - 1.64T + 7T^{2} \) |
| 11 | \( 1 - 2.94T + 11T^{2} \) |
| 13 | \( 1 - 4.94T + 13T^{2} \) |
| 17 | \( 1 + 1.28T + 17T^{2} \) |
| 19 | \( 1 + 6.81T + 19T^{2} \) |
| 23 | \( 1 - 2.22T + 23T^{2} \) |
| 29 | \( 1 + 5.66T + 29T^{2} \) |
| 37 | \( 1 + 4.22T + 37T^{2} \) |
| 41 | \( 1 - 4.81T + 41T^{2} \) |
| 43 | \( 1 - 9.51T + 43T^{2} \) |
| 47 | \( 1 + 4.45T + 47T^{2} \) |
| 53 | \( 1 + 11.1T + 53T^{2} \) |
| 59 | \( 1 - 12.5T + 59T^{2} \) |
| 61 | \( 1 + 9.74T + 61T^{2} \) |
| 67 | \( 1 - 4T + 67T^{2} \) |
| 71 | \( 1 + 16.3T + 71T^{2} \) |
| 73 | \( 1 - 8.68T + 73T^{2} \) |
| 79 | \( 1 - 3.28T + 79T^{2} \) |
| 83 | \( 1 - 7.28T + 83T^{2} \) |
| 89 | \( 1 + 6T + 89T^{2} \) |
| 97 | \( 1 - 5.87T + 97T^{2} \) |
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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
−14.19665317629604136632016168098, −13.01075003541603519926496154050, −11.38484135394109016813008487947, −10.72123019172169516969561369927, −9.319151705947122350947862216174, −8.640069728165740433677784978919, −7.74966077492167231641812230487, −6.42608229936386307467737755500, −4.08216326729393404745293124042, −1.73265192181088079091474824754,
1.73265192181088079091474824754, 4.08216326729393404745293124042, 6.42608229936386307467737755500, 7.74966077492167231641812230487, 8.640069728165740433677784978919, 9.319151705947122350947862216174, 10.72123019172169516969561369927, 11.38484135394109016813008487947, 13.01075003541603519926496154050, 14.19665317629604136632016168098
