| L(s) = 1 | + 2.05·2-s + 1.73·3-s + 2.20·4-s + 2.98·5-s + 3.55·6-s − 0.740·7-s + 0.429·8-s + 0.00697·9-s + 6.11·10-s − 5.67·11-s + 3.83·12-s + 5.82·13-s − 1.51·14-s + 5.17·15-s − 3.53·16-s − 17-s + 0.0143·18-s + 7.13·19-s + 6.58·20-s − 1.28·21-s − 11.6·22-s + 2.83·23-s + 0.744·24-s + 3.89·25-s + 11.9·26-s − 5.19·27-s − 1.63·28-s + ⋯ |
| L(s) = 1 | + 1.45·2-s + 1.00·3-s + 1.10·4-s + 1.33·5-s + 1.45·6-s − 0.279·7-s + 0.151·8-s + 0.00232·9-s + 1.93·10-s − 1.70·11-s + 1.10·12-s + 1.61·13-s − 0.405·14-s + 1.33·15-s − 0.884·16-s − 0.242·17-s + 0.00337·18-s + 1.63·19-s + 1.47·20-s − 0.280·21-s − 2.48·22-s + 0.591·23-s + 0.151·24-s + 0.778·25-s + 2.34·26-s − 0.998·27-s − 0.309·28-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
| \(L(1)\) |
\(\approx\) |
\(5.131168967\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.131168967\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 2.05T + 2T^{2} \) |
| 3 | \( 1 - 1.73T + 3T^{2} \) |
| 5 | \( 1 - 2.98T + 5T^{2} \) |
| 7 | \( 1 + 0.740T + 7T^{2} \) |
| 11 | \( 1 + 5.67T + 11T^{2} \) |
| 13 | \( 1 - 5.82T + 13T^{2} \) |
| 19 | \( 1 - 7.13T + 19T^{2} \) |
| 23 | \( 1 - 2.83T + 23T^{2} \) |
| 29 | \( 1 - 7.10T + 29T^{2} \) |
| 31 | \( 1 + 3.32T + 31T^{2} \) |
| 37 | \( 1 + 9.15T + 37T^{2} \) |
| 41 | \( 1 + 6.21T + 41T^{2} \) |
| 43 | \( 1 - 3.79T + 43T^{2} \) |
| 47 | \( 1 + 5.37T + 47T^{2} \) |
| 53 | \( 1 - 1.82T + 53T^{2} \) |
| 61 | \( 1 + 6.74T + 61T^{2} \) |
| 67 | \( 1 + 12.2T + 67T^{2} \) |
| 71 | \( 1 + 3.54T + 71T^{2} \) |
| 73 | \( 1 - 4.72T + 73T^{2} \) |
| 79 | \( 1 + 15.3T + 79T^{2} \) |
| 83 | \( 1 - 2.58T + 83T^{2} \) |
| 89 | \( 1 - 8.96T + 89T^{2} \) |
| 97 | \( 1 - 10.4T + 97T^{2} \) |
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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
−10.00331237226788481137460813655, −9.065311654642949449616072890595, −8.433214467866035813365830819958, −7.27339533743264009440027720658, −6.16982642188749118904665746815, −5.57038486800422403602410343749, −4.86068516907933123202043661854, −3.27749005724992040387459569778, −3.04101900755138541379329326380, −1.87501204322454790357780088642,
1.87501204322454790357780088642, 3.04101900755138541379329326380, 3.27749005724992040387459569778, 4.86068516907933123202043661854, 5.57038486800422403602410343749, 6.16982642188749118904665746815, 7.27339533743264009440027720658, 8.433214467866035813365830819958, 9.065311654642949449616072890595, 10.00331237226788481137460813655
