L(s) = 1 | − 2-s − 1.90·3-s + 4-s − 3.49·5-s + 1.90·6-s − 2.88·7-s − 8-s + 0.629·9-s + 3.49·10-s − 2.01·11-s − 1.90·12-s − 5.40·13-s + 2.88·14-s + 6.66·15-s + 16-s + 4.25·17-s − 0.629·18-s + 2.92·19-s − 3.49·20-s + 5.49·21-s + 2.01·22-s − 2.94·23-s + 1.90·24-s + 7.24·25-s + 5.40·26-s + 4.51·27-s − 2.88·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 1.09·3-s + 0.5·4-s − 1.56·5-s + 0.777·6-s − 1.09·7-s − 0.353·8-s + 0.209·9-s + 1.10·10-s − 0.606·11-s − 0.549·12-s − 1.50·13-s + 0.770·14-s + 1.72·15-s + 0.250·16-s + 1.03·17-s − 0.148·18-s + 0.670·19-s − 0.782·20-s + 1.19·21-s + 0.428·22-s − 0.615·23-s + 0.388·24-s + 1.44·25-s + 1.06·26-s + 0.869·27-s − 0.545·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + T \) |
| 2017 | \( 1 + T \) |
good | 3 | \( 1 + 1.90T + 3T^{2} \) |
| 5 | \( 1 + 3.49T + 5T^{2} \) |
| 7 | \( 1 + 2.88T + 7T^{2} \) |
| 11 | \( 1 + 2.01T + 11T^{2} \) |
| 13 | \( 1 + 5.40T + 13T^{2} \) |
| 17 | \( 1 - 4.25T + 17T^{2} \) |
| 19 | \( 1 - 2.92T + 19T^{2} \) |
| 23 | \( 1 + 2.94T + 23T^{2} \) |
| 29 | \( 1 + 3.50T + 29T^{2} \) |
| 31 | \( 1 + 7.26T + 31T^{2} \) |
| 37 | \( 1 - 7.06T + 37T^{2} \) |
| 41 | \( 1 + 2.34T + 41T^{2} \) |
| 43 | \( 1 - 4.99T + 43T^{2} \) |
| 47 | \( 1 - 8.87T + 47T^{2} \) |
| 53 | \( 1 - 7.95T + 53T^{2} \) |
| 59 | \( 1 - 11.9T + 59T^{2} \) |
| 61 | \( 1 + 0.563T + 61T^{2} \) |
| 67 | \( 1 + 0.351T + 67T^{2} \) |
| 71 | \( 1 + 4.94T + 71T^{2} \) |
| 73 | \( 1 + 1.76T + 73T^{2} \) |
| 79 | \( 1 + 4.78T + 79T^{2} \) |
| 83 | \( 1 + 7.48T + 83T^{2} \) |
| 89 | \( 1 - 12.6T + 89T^{2} \) |
| 97 | \( 1 - 0.00634T + 97T^{2} \) |
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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
−7.77492706371925939263596550210, −7.45621724354734482653715868720, −6.87042966289074055217965076774, −5.78604306180625219091846176812, −5.32838243621026401103625812670, −4.23510327956857365599242514548, −3.36748189747382611157344955604, −2.54107702928981780944954387491, −0.69967277835317185102181169026, 0,
0.69967277835317185102181169026, 2.54107702928981780944954387491, 3.36748189747382611157344955604, 4.23510327956857365599242514548, 5.32838243621026401103625812670, 5.78604306180625219091846176812, 6.87042966289074055217965076774, 7.45621724354734482653715868720, 7.77492706371925939263596550210