| L(s) = 1 | + 5.38·2-s + 6.80·3-s + 20.9·4-s + 6.71·5-s + 36.6·6-s + 6.75·7-s + 69.9·8-s + 19.3·9-s + 36.1·10-s − 8.82·11-s + 142.·12-s − 71.2·13-s + 36.3·14-s + 45.7·15-s + 208.·16-s − 11.3·17-s + 104.·18-s − 37.4·19-s + 140.·20-s + 45.9·21-s − 47.5·22-s − 23·23-s + 476.·24-s − 79.9·25-s − 383.·26-s − 52.0·27-s + 141.·28-s + ⋯ |
| L(s) = 1 | + 1.90·2-s + 1.31·3-s + 2.62·4-s + 0.600·5-s + 2.49·6-s + 0.364·7-s + 3.09·8-s + 0.716·9-s + 1.14·10-s − 0.241·11-s + 3.43·12-s − 1.51·13-s + 0.694·14-s + 0.786·15-s + 3.26·16-s − 0.162·17-s + 1.36·18-s − 0.452·19-s + 1.57·20-s + 0.477·21-s − 0.460·22-s − 0.208·23-s + 4.05·24-s − 0.639·25-s − 2.89·26-s − 0.370·27-s + 0.956·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 667 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 667 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(11.16915755\) |
| \(L(\frac12)\) |
\(\approx\) |
\(11.16915755\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 23 | \( 1 + 23T \) |
| 29 | \( 1 + 29T \) |
| good | 2 | \( 1 - 5.38T + 8T^{2} \) |
| 3 | \( 1 - 6.80T + 27T^{2} \) |
| 5 | \( 1 - 6.71T + 125T^{2} \) |
| 7 | \( 1 - 6.75T + 343T^{2} \) |
| 11 | \( 1 + 8.82T + 1.33e3T^{2} \) |
| 13 | \( 1 + 71.2T + 2.19e3T^{2} \) |
| 17 | \( 1 + 11.3T + 4.91e3T^{2} \) |
| 19 | \( 1 + 37.4T + 6.85e3T^{2} \) |
| 31 | \( 1 - 15.4T + 2.97e4T^{2} \) |
| 37 | \( 1 + 19.5T + 5.06e4T^{2} \) |
| 41 | \( 1 - 194.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 187.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 99.3T + 1.03e5T^{2} \) |
| 53 | \( 1 + 106.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 40.2T + 2.05e5T^{2} \) |
| 61 | \( 1 - 413.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 164.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 143.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 16.7T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.02e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 308.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.22e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 939.T + 9.12e5T^{2} \) |
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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
−10.20692521148045356063439437025, −9.353684913940847233309263919546, −8.014791753504785226417884919343, −7.39264747360128812744502506121, −6.33494548007969605553389262718, −5.33039174159245365691758723466, −4.50903931010472412293516012784, −3.50810373747999258753327430934, −2.44890626336489224509195597047, −2.02609079510467699340664209218,
2.02609079510467699340664209218, 2.44890626336489224509195597047, 3.50810373747999258753327430934, 4.50903931010472412293516012784, 5.33039174159245365691758723466, 6.33494548007969605553389262718, 7.39264747360128812744502506121, 8.014791753504785226417884919343, 9.353684913940847233309263919546, 10.20692521148045356063439437025
