| L(s) = 1 | − 1.64·2-s + 2.69·3-s + 0.717·4-s − 2.78·5-s − 4.43·6-s − 0.853·7-s + 2.11·8-s + 4.24·9-s + 4.59·10-s − 2.80·11-s + 1.93·12-s + 1.03·13-s + 1.40·14-s − 7.50·15-s − 4.92·16-s + 1.76·17-s − 6.99·18-s − 5.56·19-s − 2.00·20-s − 2.29·21-s + 4.62·22-s + 4.11·23-s + 5.68·24-s + 2.77·25-s − 1.69·26-s + 3.34·27-s − 0.612·28-s + ⋯ |
| L(s) = 1 | − 1.16·2-s + 1.55·3-s + 0.358·4-s − 1.24·5-s − 1.81·6-s − 0.322·7-s + 0.747·8-s + 1.41·9-s + 1.45·10-s − 0.844·11-s + 0.557·12-s + 0.285·13-s + 0.376·14-s − 1.93·15-s − 1.23·16-s + 0.427·17-s − 1.64·18-s − 1.27·19-s − 0.447·20-s − 0.501·21-s + 0.985·22-s + 0.857·23-s + 1.16·24-s + 0.555·25-s − 0.333·26-s + 0.642·27-s − 0.115·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 983 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 983 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 983 | \( 1 + T \) |
| good | 2 | \( 1 + 1.64T + 2T^{2} \) |
| 3 | \( 1 - 2.69T + 3T^{2} \) |
| 5 | \( 1 + 2.78T + 5T^{2} \) |
| 7 | \( 1 + 0.853T + 7T^{2} \) |
| 11 | \( 1 + 2.80T + 11T^{2} \) |
| 13 | \( 1 - 1.03T + 13T^{2} \) |
| 17 | \( 1 - 1.76T + 17T^{2} \) |
| 19 | \( 1 + 5.56T + 19T^{2} \) |
| 23 | \( 1 - 4.11T + 23T^{2} \) |
| 29 | \( 1 - 2.86T + 29T^{2} \) |
| 31 | \( 1 + 6.42T + 31T^{2} \) |
| 37 | \( 1 + 8.71T + 37T^{2} \) |
| 41 | \( 1 + 10.7T + 41T^{2} \) |
| 43 | \( 1 + 2.20T + 43T^{2} \) |
| 47 | \( 1 - 1.85T + 47T^{2} \) |
| 53 | \( 1 - 7.10T + 53T^{2} \) |
| 59 | \( 1 + 6.33T + 59T^{2} \) |
| 61 | \( 1 + 12.7T + 61T^{2} \) |
| 67 | \( 1 + 0.335T + 67T^{2} \) |
| 71 | \( 1 + 5.42T + 71T^{2} \) |
| 73 | \( 1 + 0.891T + 73T^{2} \) |
| 79 | \( 1 - 6.71T + 79T^{2} \) |
| 83 | \( 1 - 13.3T + 83T^{2} \) |
| 89 | \( 1 + 3.65T + 89T^{2} \) |
| 97 | \( 1 - 4.56T + 97T^{2} \) |
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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
−9.199383617251421107004909296678, −8.712772162664366823924008530485, −8.085357116479711171812991847978, −7.58294342453894823729422590545, −6.78384274847623873038145890623, −4.92625847740687601214675968657, −3.86646071939118288817289191857, −3.09539910004198592785864316745, −1.80483175474620703806645362295, 0,
1.80483175474620703806645362295, 3.09539910004198592785864316745, 3.86646071939118288817289191857, 4.92625847740687601214675968657, 6.78384274847623873038145890623, 7.58294342453894823729422590545, 8.085357116479711171812991847978, 8.712772162664366823924008530485, 9.199383617251421107004909296678
