| L(s) = 1 | − 0.256·2-s + 3-s − 1.93·4-s − 0.330·5-s − 0.256·6-s + 1.25·7-s + 1.00·8-s + 9-s + 0.0848·10-s − 0.708·11-s − 1.93·12-s − 13-s − 0.321·14-s − 0.330·15-s + 3.60·16-s − 5.11·17-s − 0.256·18-s + 4.61·19-s + 0.639·20-s + 1.25·21-s + 0.181·22-s + 4.94·23-s + 1.00·24-s − 4.89·25-s + 0.256·26-s + 27-s − 2.42·28-s + ⋯ |
| L(s) = 1 | − 0.181·2-s + 0.577·3-s − 0.967·4-s − 0.147·5-s − 0.104·6-s + 0.473·7-s + 0.356·8-s + 0.333·9-s + 0.0268·10-s − 0.213·11-s − 0.558·12-s − 0.277·13-s − 0.0858·14-s − 0.0853·15-s + 0.902·16-s − 1.24·17-s − 0.0604·18-s + 1.05·19-s + 0.143·20-s + 0.273·21-s + 0.0387·22-s + 1.03·23-s + 0.205·24-s − 0.978·25-s + 0.0502·26-s + 0.192·27-s − 0.457·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| 103 | \( 1 - T \) |
| good | 2 | \( 1 + 0.256T + 2T^{2} \) |
| 5 | \( 1 + 0.330T + 5T^{2} \) |
| 7 | \( 1 - 1.25T + 7T^{2} \) |
| 11 | \( 1 + 0.708T + 11T^{2} \) |
| 17 | \( 1 + 5.11T + 17T^{2} \) |
| 19 | \( 1 - 4.61T + 19T^{2} \) |
| 23 | \( 1 - 4.94T + 23T^{2} \) |
| 29 | \( 1 + 5.42T + 29T^{2} \) |
| 31 | \( 1 + 7.69T + 31T^{2} \) |
| 37 | \( 1 + 10.3T + 37T^{2} \) |
| 41 | \( 1 - 8.14T + 41T^{2} \) |
| 43 | \( 1 - 6.55T + 43T^{2} \) |
| 47 | \( 1 + 6.08T + 47T^{2} \) |
| 53 | \( 1 - 6.75T + 53T^{2} \) |
| 59 | \( 1 - 13.2T + 59T^{2} \) |
| 61 | \( 1 - 0.799T + 61T^{2} \) |
| 67 | \( 1 + 2.78T + 67T^{2} \) |
| 71 | \( 1 + 4.43T + 71T^{2} \) |
| 73 | \( 1 - 14.2T + 73T^{2} \) |
| 79 | \( 1 + 0.500T + 79T^{2} \) |
| 83 | \( 1 + 5.34T + 83T^{2} \) |
| 89 | \( 1 + 10.3T + 89T^{2} \) |
| 97 | \( 1 + 9.27T + 97T^{2} \) |
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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
−8.144579227833899508955043938305, −7.49862554008687543175120823321, −6.92110320866709816036448462870, −5.53666198842131369441724724516, −5.11659880615669817998914159097, −4.14164561549096217219329901749, −3.59298086439829525040572226506, −2.43868589067931884299178195323, −1.40358393588057097169111600984, 0,
1.40358393588057097169111600984, 2.43868589067931884299178195323, 3.59298086439829525040572226506, 4.14164561549096217219329901749, 5.11659880615669817998914159097, 5.53666198842131369441724724516, 6.92110320866709816036448462870, 7.49862554008687543175120823321, 8.144579227833899508955043938305
