| L(s) = 1 | − 2.28·2-s + 0.0353·3-s + 3.21·4-s − 0.992·5-s − 0.0808·6-s − 0.288·7-s − 2.78·8-s − 2.99·9-s + 2.26·10-s + 2.40·11-s + 0.113·12-s + 0.404·13-s + 0.659·14-s − 0.0351·15-s − 0.0728·16-s − 17-s + 6.85·18-s − 3.02·19-s − 3.19·20-s − 0.0102·21-s − 5.49·22-s + 7.26·23-s − 0.0985·24-s − 4.01·25-s − 0.924·26-s − 0.212·27-s − 0.928·28-s + ⋯ |
| L(s) = 1 | − 1.61·2-s + 0.0204·3-s + 1.60·4-s − 0.443·5-s − 0.0329·6-s − 0.109·7-s − 0.985·8-s − 0.999·9-s + 0.717·10-s + 0.725·11-s + 0.0328·12-s + 0.112·13-s + 0.176·14-s − 0.00906·15-s − 0.0182·16-s − 0.242·17-s + 1.61·18-s − 0.694·19-s − 0.714·20-s − 0.00222·21-s − 1.17·22-s + 1.51·23-s − 0.0201·24-s − 0.802·25-s − 0.181·26-s − 0.0408·27-s − 0.175·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1003 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5407212790\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5407212790\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 + T \) |
| 59 | \( 1 - T \) |
| good | 2 | \( 1 + 2.28T + 2T^{2} \) |
| 3 | \( 1 - 0.0353T + 3T^{2} \) |
| 5 | \( 1 + 0.992T + 5T^{2} \) |
| 7 | \( 1 + 0.288T + 7T^{2} \) |
| 11 | \( 1 - 2.40T + 11T^{2} \) |
| 13 | \( 1 - 0.404T + 13T^{2} \) |
| 19 | \( 1 + 3.02T + 19T^{2} \) |
| 23 | \( 1 - 7.26T + 23T^{2} \) |
| 29 | \( 1 - 4.41T + 29T^{2} \) |
| 31 | \( 1 - 0.846T + 31T^{2} \) |
| 37 | \( 1 + 6.66T + 37T^{2} \) |
| 41 | \( 1 - 8.24T + 41T^{2} \) |
| 43 | \( 1 - 7.83T + 43T^{2} \) |
| 47 | \( 1 + 5.05T + 47T^{2} \) |
| 53 | \( 1 + 4.17T + 53T^{2} \) |
| 61 | \( 1 - 8.06T + 61T^{2} \) |
| 67 | \( 1 - 1.30T + 67T^{2} \) |
| 71 | \( 1 - 7.17T + 71T^{2} \) |
| 73 | \( 1 + 1.04T + 73T^{2} \) |
| 79 | \( 1 + 0.314T + 79T^{2} \) |
| 83 | \( 1 - 15.2T + 83T^{2} \) |
| 89 | \( 1 - 3.40T + 89T^{2} \) |
| 97 | \( 1 - 12.7T + 97T^{2} \) |
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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
−9.750298083786118553516507165360, −9.008534068608561160303605148758, −8.517816822247815280268767701697, −7.73694793669097173268861838471, −6.83110177762087531124419410119, −6.10405708559050001351541477740, −4.70501947356886245562398179092, −3.37487786553801286571242940285, −2.17362237863045583663123540660, −0.71463996128510246576589916925,
0.71463996128510246576589916925, 2.17362237863045583663123540660, 3.37487786553801286571242940285, 4.70501947356886245562398179092, 6.10405708559050001351541477740, 6.83110177762087531124419410119, 7.73694793669097173268861838471, 8.517816822247815280268767701697, 9.008534068608561160303605148758, 9.750298083786118553516507165360
