| L(s) = 1 | + 1.72·2-s − 2.82·3-s + 0.965·4-s − 5-s − 4.85·6-s + 1.90·7-s − 1.78·8-s + 4.96·9-s − 1.72·10-s − 0.406·11-s − 2.72·12-s + 0.826·13-s + 3.27·14-s + 2.82·15-s − 4.99·16-s − 1.05·17-s + 8.54·18-s + 6.14·19-s − 0.965·20-s − 5.37·21-s − 0.699·22-s + 7.03·23-s + 5.02·24-s + 25-s + 1.42·26-s − 5.54·27-s + 1.83·28-s + ⋯ |
| L(s) = 1 | + 1.21·2-s − 1.62·3-s + 0.482·4-s − 0.447·5-s − 1.98·6-s + 0.719·7-s − 0.629·8-s + 1.65·9-s − 0.544·10-s − 0.122·11-s − 0.786·12-s + 0.229·13-s + 0.876·14-s + 0.728·15-s − 1.24·16-s − 0.255·17-s + 2.01·18-s + 1.40·19-s − 0.215·20-s − 1.17·21-s − 0.149·22-s + 1.46·23-s + 1.02·24-s + 0.200·25-s + 0.279·26-s − 1.06·27-s + 0.347·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2005 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2005 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + T \) |
| 401 | \( 1 + T \) |
| good | 2 | \( 1 - 1.72T + 2T^{2} \) |
| 3 | \( 1 + 2.82T + 3T^{2} \) |
| 7 | \( 1 - 1.90T + 7T^{2} \) |
| 11 | \( 1 + 0.406T + 11T^{2} \) |
| 13 | \( 1 - 0.826T + 13T^{2} \) |
| 17 | \( 1 + 1.05T + 17T^{2} \) |
| 19 | \( 1 - 6.14T + 19T^{2} \) |
| 23 | \( 1 - 7.03T + 23T^{2} \) |
| 29 | \( 1 + 10.0T + 29T^{2} \) |
| 31 | \( 1 + 10.3T + 31T^{2} \) |
| 37 | \( 1 + 6.70T + 37T^{2} \) |
| 41 | \( 1 - 9.72T + 41T^{2} \) |
| 43 | \( 1 + 4.37T + 43T^{2} \) |
| 47 | \( 1 + 8.18T + 47T^{2} \) |
| 53 | \( 1 + 0.699T + 53T^{2} \) |
| 59 | \( 1 - 2.39T + 59T^{2} \) |
| 61 | \( 1 + 8.98T + 61T^{2} \) |
| 67 | \( 1 + 6.60T + 67T^{2} \) |
| 71 | \( 1 - 0.219T + 71T^{2} \) |
| 73 | \( 1 - 7.26T + 73T^{2} \) |
| 79 | \( 1 + 7.60T + 79T^{2} \) |
| 83 | \( 1 + 7.58T + 83T^{2} \) |
| 89 | \( 1 + 7.39T + 89T^{2} \) |
| 97 | \( 1 - 4.54T + 97T^{2} \) |
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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
−8.855043447476931574070601028959, −7.52992076835331809674186793922, −7.01234635908080354200316014238, −6.01045134704992044967688999987, −5.26385951398081226041118671389, −5.04456505128076772129569146845, −4.06406093804033461269222013477, −3.20996992482954028920153226414, −1.50622809690269636644688877925, 0,
1.50622809690269636644688877925, 3.20996992482954028920153226414, 4.06406093804033461269222013477, 5.04456505128076772129569146845, 5.26385951398081226041118671389, 6.01045134704992044967688999987, 7.01234635908080354200316014238, 7.52992076835331809674186793922, 8.855043447476931574070601028959
