L(s) = 1 | − 2.36·2-s − 1.50·3-s + 3.60·4-s − 1.85·5-s + 3.55·6-s + 0.0863·7-s − 3.79·8-s − 0.747·9-s + 4.40·10-s − 2.66·11-s − 5.40·12-s + 3.74·13-s − 0.204·14-s + 2.79·15-s + 1.77·16-s − 6.18·17-s + 1.76·18-s + 2.13·19-s − 6.69·20-s − 0.129·21-s + 6.30·22-s + 0.868·23-s + 5.69·24-s − 1.54·25-s − 8.85·26-s + 5.62·27-s + 0.310·28-s + ⋯ |
L(s) = 1 | − 1.67·2-s − 0.866·3-s + 1.80·4-s − 0.831·5-s + 1.45·6-s + 0.0326·7-s − 1.34·8-s − 0.249·9-s + 1.39·10-s − 0.802·11-s − 1.56·12-s + 1.03·13-s − 0.0546·14-s + 0.720·15-s + 0.442·16-s − 1.49·17-s + 0.417·18-s + 0.490·19-s − 1.49·20-s − 0.0282·21-s + 1.34·22-s + 0.181·23-s + 1.16·24-s − 0.308·25-s − 1.73·26-s + 1.08·27-s + 0.0587·28-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
\(L(1)\) |
\(\approx\) |
\(0.2148983647\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2148983647\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.36T + 2T^{2} \) |
| 3 | \( 1 + 1.50T + 3T^{2} \) |
| 5 | \( 1 + 1.85T + 5T^{2} \) |
| 7 | \( 1 - 0.0863T + 7T^{2} \) |
| 11 | \( 1 + 2.66T + 11T^{2} \) |
| 13 | \( 1 - 3.74T + 13T^{2} \) |
| 17 | \( 1 + 6.18T + 17T^{2} \) |
| 19 | \( 1 - 2.13T + 19T^{2} \) |
| 23 | \( 1 - 0.868T + 23T^{2} \) |
| 29 | \( 1 + 6.22T + 29T^{2} \) |
| 31 | \( 1 + 3.34T + 31T^{2} \) |
| 37 | \( 1 + 4.13T + 37T^{2} \) |
| 41 | \( 1 + 2.53T + 41T^{2} \) |
| 43 | \( 1 + 4.06T + 43T^{2} \) |
| 47 | \( 1 + 1.46T + 47T^{2} \) |
| 53 | \( 1 - 13.2T + 53T^{2} \) |
| 59 | \( 1 + 13.2T + 59T^{2} \) |
| 61 | \( 1 - 14.6T + 61T^{2} \) |
| 67 | \( 1 + 5.47T + 67T^{2} \) |
| 71 | \( 1 - 13.7T + 71T^{2} \) |
| 73 | \( 1 + 0.172T + 73T^{2} \) |
| 79 | \( 1 + 0.692T + 79T^{2} \) |
| 83 | \( 1 - 3.02T + 83T^{2} \) |
| 89 | \( 1 - 10.4T + 89T^{2} \) |
| 97 | \( 1 - 9.80T + 97T^{2} \) |
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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
−10.07464149302097663105435125873, −8.971245636298411801170108822161, −8.477190216194691759129960445322, −7.64612724321614183766781006801, −6.87658714250480789781375544959, −5.99842963991626446822193141396, −4.90857116191049102533823169725, −3.51198896840123187216389899876, −2.02636900038263944441569331114, −0.46412062030612892329235839690,
0.46412062030612892329235839690, 2.02636900038263944441569331114, 3.51198896840123187216389899876, 4.90857116191049102533823169725, 5.99842963991626446822193141396, 6.87658714250480789781375544959, 7.64612724321614183766781006801, 8.477190216194691759129960445322, 8.971245636298411801170108822161, 10.07464149302097663105435125873