| L(s) = 1 | + 2.33·2-s − 1.13·3-s + 3.43·4-s + 1.77·5-s − 2.65·6-s + 5.24·7-s + 3.35·8-s − 1.70·9-s + 4.14·10-s − 5.08·11-s − 3.90·12-s + 5.60·13-s + 12.2·14-s − 2.01·15-s + 0.948·16-s − 2.45·17-s − 3.98·18-s + 6.33·19-s + 6.10·20-s − 5.96·21-s − 11.8·22-s + 0.354·23-s − 3.81·24-s − 1.84·25-s + 13.0·26-s + 5.35·27-s + 18.0·28-s + ⋯ |
| L(s) = 1 | + 1.64·2-s − 0.656·3-s + 1.71·4-s + 0.794·5-s − 1.08·6-s + 1.98·7-s + 1.18·8-s − 0.569·9-s + 1.30·10-s − 1.53·11-s − 1.12·12-s + 1.55·13-s + 3.27·14-s − 0.521·15-s + 0.237·16-s − 0.595·17-s − 0.939·18-s + 1.45·19-s + 1.36·20-s − 1.30·21-s − 2.52·22-s + 0.0738·23-s − 0.778·24-s − 0.369·25-s + 2.56·26-s + 1.02·27-s + 3.41·28-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
| \(L(1)\) |
\(\approx\) |
\(4.125870811\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.125870811\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 2.33T + 2T^{2} \) |
| 3 | \( 1 + 1.13T + 3T^{2} \) |
| 5 | \( 1 - 1.77T + 5T^{2} \) |
| 7 | \( 1 - 5.24T + 7T^{2} \) |
| 11 | \( 1 + 5.08T + 11T^{2} \) |
| 13 | \( 1 - 5.60T + 13T^{2} \) |
| 17 | \( 1 + 2.45T + 17T^{2} \) |
| 19 | \( 1 - 6.33T + 19T^{2} \) |
| 23 | \( 1 - 0.354T + 23T^{2} \) |
| 29 | \( 1 + 4.25T + 29T^{2} \) |
| 31 | \( 1 + 8.81T + 31T^{2} \) |
| 37 | \( 1 + 1.48T + 37T^{2} \) |
| 41 | \( 1 - 2.18T + 41T^{2} \) |
| 43 | \( 1 - 7.05T + 43T^{2} \) |
| 47 | \( 1 - 3.87T + 47T^{2} \) |
| 53 | \( 1 + 5.95T + 53T^{2} \) |
| 59 | \( 1 + 4.91T + 59T^{2} \) |
| 61 | \( 1 - 6.30T + 61T^{2} \) |
| 67 | \( 1 + 7.63T + 67T^{2} \) |
| 71 | \( 1 - 2.79T + 71T^{2} \) |
| 73 | \( 1 - 12.0T + 73T^{2} \) |
| 79 | \( 1 - 1.79T + 79T^{2} \) |
| 83 | \( 1 + 17.2T + 83T^{2} \) |
| 89 | \( 1 + 14.4T + 89T^{2} \) |
| 97 | \( 1 + 13.5T + 97T^{2} \) |
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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
−10.66856104557993770494876673372, −9.102492006200695696560905105693, −8.105615070250899456363083358909, −7.26285622441974335032759563699, −5.89291757187797611543723214032, −5.51180238348959478805593036565, −5.08451819572063409936517361102, −3.98724786835396367877216852923, −2.67688869089907328118931620589, −1.63505757934576342740746586620,
1.63505757934576342740746586620, 2.67688869089907328118931620589, 3.98724786835396367877216852923, 5.08451819572063409936517361102, 5.51180238348959478805593036565, 5.89291757187797611543723214032, 7.26285622441974335032759563699, 8.105615070250899456363083358909, 9.102492006200695696560905105693, 10.66856104557993770494876673372
