| L(s) = 1 | − 2.74·2-s − 0.794·3-s + 5.52·4-s − 3.63·5-s + 2.17·6-s − 4.59·7-s − 9.68·8-s − 2.36·9-s + 9.98·10-s + 3.75·11-s − 4.39·12-s + 5.08·13-s + 12.6·14-s + 2.89·15-s + 15.5·16-s + 17-s + 6.50·18-s + 6.51·19-s − 20.1·20-s + 3.65·21-s − 10.3·22-s + 1.37·23-s + 7.69·24-s + 8.24·25-s − 13.9·26-s + 4.26·27-s − 25.4·28-s + ⋯ |
| L(s) = 1 | − 1.94·2-s − 0.458·3-s + 2.76·4-s − 1.62·5-s + 0.889·6-s − 1.73·7-s − 3.42·8-s − 0.789·9-s + 3.15·10-s + 1.13·11-s − 1.26·12-s + 1.41·13-s + 3.37·14-s + 0.746·15-s + 3.87·16-s + 0.242·17-s + 1.53·18-s + 1.49·19-s − 4.50·20-s + 0.796·21-s − 2.19·22-s + 0.285·23-s + 1.57·24-s + 1.64·25-s − 2.73·26-s + 0.820·27-s − 4.80·28-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
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 - T \) |
| 59 | \( 1 - T \) |
| good | 2 | \( 1 + 2.74T + 2T^{2} \) |
| 3 | \( 1 + 0.794T + 3T^{2} \) |
| 5 | \( 1 + 3.63T + 5T^{2} \) |
| 7 | \( 1 + 4.59T + 7T^{2} \) |
| 11 | \( 1 - 3.75T + 11T^{2} \) |
| 13 | \( 1 - 5.08T + 13T^{2} \) |
| 19 | \( 1 - 6.51T + 19T^{2} \) |
| 23 | \( 1 - 1.37T + 23T^{2} \) |
| 29 | \( 1 + 2.06T + 29T^{2} \) |
| 31 | \( 1 + 3.21T + 31T^{2} \) |
| 37 | \( 1 - 2.18T + 37T^{2} \) |
| 41 | \( 1 + 5.62T + 41T^{2} \) |
| 43 | \( 1 + 8.67T + 43T^{2} \) |
| 47 | \( 1 + 7.91T + 47T^{2} \) |
| 53 | \( 1 - 2.24T + 53T^{2} \) |
| 61 | \( 1 - 4.17T + 61T^{2} \) |
| 67 | \( 1 - 7.70T + 67T^{2} \) |
| 71 | \( 1 + 7.33T + 71T^{2} \) |
| 73 | \( 1 + 12.9T + 73T^{2} \) |
| 79 | \( 1 - 0.0406T + 79T^{2} \) |
| 83 | \( 1 - 1.31T + 83T^{2} \) |
| 89 | \( 1 - 5.21T + 89T^{2} \) |
| 97 | \( 1 + 4.44T + 97T^{2} \) |
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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
−9.394936338740780233404716331692, −8.787072938988482995929032844233, −8.106827973516999339456890359952, −7.09536190911729472193331208112, −6.60911703492313862979186431881, −5.77801124463672270474819828152, −3.47864610629667978342882898507, −3.26324165716789478590745443506, −1.05375389875380089079224032697, 0,
1.05375389875380089079224032697, 3.26324165716789478590745443506, 3.47864610629667978342882898507, 5.77801124463672270474819828152, 6.60911703492313862979186431881, 7.09536190911729472193331208112, 8.106827973516999339456890359952, 8.787072938988482995929032844233, 9.394936338740780233404716331692
