| L(s) = 1 | + 1.01·2-s − 3.15·3-s − 0.969·4-s − 2.87·5-s − 3.19·6-s − 2.07·7-s − 3.01·8-s + 6.93·9-s − 2.91·10-s − 5.58·11-s + 3.05·12-s − 2.86·13-s − 2.10·14-s + 9.05·15-s − 1.12·16-s − 6.40·17-s + 7.04·18-s − 4.17·19-s + 2.78·20-s + 6.55·21-s − 5.66·22-s + 9.05·23-s + 9.50·24-s + 3.25·25-s − 2.90·26-s − 12.4·27-s + 2.01·28-s + ⋯ |
| L(s) = 1 | + 0.717·2-s − 1.81·3-s − 0.484·4-s − 1.28·5-s − 1.30·6-s − 0.785·7-s − 1.06·8-s + 2.31·9-s − 0.922·10-s − 1.68·11-s + 0.882·12-s − 0.794·13-s − 0.563·14-s + 2.33·15-s − 0.280·16-s − 1.55·17-s + 1.65·18-s − 0.958·19-s + 0.622·20-s + 1.42·21-s − 1.20·22-s + 1.88·23-s + 1.93·24-s + 0.650·25-s − 0.569·26-s − 2.38·27-s + 0.380·28-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
| \(L(1)\) |
\(\approx\) |
\(0.02879994845\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.02879994845\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 1.01T + 2T^{2} \) |
| 3 | \( 1 + 3.15T + 3T^{2} \) |
| 5 | \( 1 + 2.87T + 5T^{2} \) |
| 7 | \( 1 + 2.07T + 7T^{2} \) |
| 11 | \( 1 + 5.58T + 11T^{2} \) |
| 13 | \( 1 + 2.86T + 13T^{2} \) |
| 17 | \( 1 + 6.40T + 17T^{2} \) |
| 19 | \( 1 + 4.17T + 19T^{2} \) |
| 23 | \( 1 - 9.05T + 23T^{2} \) |
| 29 | \( 1 - 1.31T + 29T^{2} \) |
| 31 | \( 1 + 4.51T + 31T^{2} \) |
| 37 | \( 1 - 4.87T + 37T^{2} \) |
| 41 | \( 1 + 5.42T + 41T^{2} \) |
| 43 | \( 1 + 3.82T + 43T^{2} \) |
| 47 | \( 1 - 0.0396T + 47T^{2} \) |
| 53 | \( 1 + 6.55T + 53T^{2} \) |
| 59 | \( 1 + 7.60T + 59T^{2} \) |
| 61 | \( 1 - 8.37T + 61T^{2} \) |
| 67 | \( 1 + 2.97T + 67T^{2} \) |
| 71 | \( 1 + 5.05T + 71T^{2} \) |
| 73 | \( 1 - 15.5T + 73T^{2} \) |
| 79 | \( 1 + 14.5T + 79T^{2} \) |
| 83 | \( 1 + 11.7T + 83T^{2} \) |
| 89 | \( 1 + 7.02T + 89T^{2} \) |
| 97 | \( 1 - 9.64T + 97T^{2} \) |
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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
−10.33667890435366800620874007098, −9.321219554055377430077895444674, −8.182516488173477090490201458868, −7.08743535152257616790038637857, −6.50637246430521998088413494861, −5.38175079573136981823794386569, −4.81182365479198687283698528588, −4.18396508830027480720632969508, −2.89414831798372170936054503637, −0.12112312657235991986107081630,
0.12112312657235991986107081630, 2.89414831798372170936054503637, 4.18396508830027480720632969508, 4.81182365479198687283698528588, 5.38175079573136981823794386569, 6.50637246430521998088413494861, 7.08743535152257616790038637857, 8.182516488173477090490201458868, 9.321219554055377430077895444674, 10.33667890435366800620874007098
