| L(s) = 1 | − 2·2-s − 6.79·3-s + 4·4-s + 13.5·6-s − 8·8-s + 19.1·9-s + 69.2·11-s − 27.1·12-s + 28.3·13-s + 16·16-s − 11.3·17-s − 38.2·18-s + 74.2·19-s − 138.·22-s − 66.6·23-s + 54.3·24-s − 56.7·26-s + 53.4·27-s − 134.·29-s − 160.·31-s − 32·32-s − 470.·33-s + 22.6·34-s + 76.5·36-s − 284.·37-s − 148.·38-s − 192.·39-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.30·3-s + 0.5·4-s + 0.924·6-s − 0.353·8-s + 0.708·9-s + 1.89·11-s − 0.653·12-s + 0.605·13-s + 0.250·16-s − 0.161·17-s − 0.500·18-s + 0.896·19-s − 1.34·22-s − 0.604·23-s + 0.462·24-s − 0.428·26-s + 0.381·27-s − 0.861·29-s − 0.930·31-s − 0.176·32-s − 2.47·33-s + 0.114·34-s + 0.354·36-s − 1.26·37-s − 0.633·38-s − 0.791·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2450 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 2T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + 6.79T + 27T^{2} \) |
| 11 | \( 1 - 69.2T + 1.33e3T^{2} \) |
| 13 | \( 1 - 28.3T + 2.19e3T^{2} \) |
| 17 | \( 1 + 11.3T + 4.91e3T^{2} \) |
| 19 | \( 1 - 74.2T + 6.85e3T^{2} \) |
| 23 | \( 1 + 66.6T + 1.21e4T^{2} \) |
| 29 | \( 1 + 134.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 160.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 284.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 149.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 513.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 411.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 354.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 210.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 587.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 1.03e3T + 3.00e5T^{2} \) |
| 71 | \( 1 + 1.03e3T + 3.57e5T^{2} \) |
| 73 | \( 1 + 743.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 609.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.14e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 735.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 333.T + 9.12e5T^{2} \) |
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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
−8.355397850334096706975344889533, −7.15925212927251918844012344785, −6.74641080492979218545101837328, −5.93314985766457207236299221851, −5.37644885278085414415286737422, −4.17104420908227547857018595684, −3.40181106749431932054457678692, −1.77247642524575325967512148099, −1.05753831400955239882615560560, 0,
1.05753831400955239882615560560, 1.77247642524575325967512148099, 3.40181106749431932054457678692, 4.17104420908227547857018595684, 5.37644885278085414415286737422, 5.93314985766457207236299221851, 6.74641080492979218545101837328, 7.15925212927251918844012344785, 8.355397850334096706975344889533
