| L(s) = 1 | + 1.24·2-s − 0.282·3-s − 0.456·4-s + 1.68·5-s − 0.350·6-s + 0.816·7-s − 3.05·8-s − 2.92·9-s + 2.09·10-s + 4.34·11-s + 0.128·12-s − 0.194·13-s + 1.01·14-s − 0.475·15-s − 2.87·16-s + 8.01·17-s − 3.62·18-s + 4.57·19-s − 0.769·20-s − 0.230·21-s + 5.40·22-s + 5.27·23-s + 0.860·24-s − 2.15·25-s − 0.241·26-s + 1.67·27-s − 0.372·28-s + ⋯ |
| L(s) = 1 | + 0.878·2-s − 0.162·3-s − 0.228·4-s + 0.753·5-s − 0.143·6-s + 0.308·7-s − 1.07·8-s − 0.973·9-s + 0.662·10-s + 1.31·11-s + 0.0371·12-s − 0.0539·13-s + 0.271·14-s − 0.122·15-s − 0.719·16-s + 1.94·17-s − 0.855·18-s + 1.04·19-s − 0.171·20-s − 0.0502·21-s + 1.15·22-s + 1.09·23-s + 0.175·24-s − 0.431·25-s − 0.0473·26-s + 0.321·27-s − 0.0703·28-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
| \(L(1)\) |
\(\approx\) |
\(2.447504088\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.447504088\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 1.24T + 2T^{2} \) |
| 3 | \( 1 + 0.282T + 3T^{2} \) |
| 5 | \( 1 - 1.68T + 5T^{2} \) |
| 7 | \( 1 - 0.816T + 7T^{2} \) |
| 11 | \( 1 - 4.34T + 11T^{2} \) |
| 13 | \( 1 + 0.194T + 13T^{2} \) |
| 17 | \( 1 - 8.01T + 17T^{2} \) |
| 19 | \( 1 - 4.57T + 19T^{2} \) |
| 23 | \( 1 - 5.27T + 23T^{2} \) |
| 29 | \( 1 + 2.24T + 29T^{2} \) |
| 31 | \( 1 - 0.0347T + 31T^{2} \) |
| 37 | \( 1 + 0.365T + 37T^{2} \) |
| 41 | \( 1 - 8.92T + 41T^{2} \) |
| 43 | \( 1 + 1.84T + 43T^{2} \) |
| 47 | \( 1 - 5.33T + 47T^{2} \) |
| 53 | \( 1 + 8.80T + 53T^{2} \) |
| 59 | \( 1 - 0.487T + 59T^{2} \) |
| 61 | \( 1 - 4.72T + 61T^{2} \) |
| 67 | \( 1 + 10.1T + 67T^{2} \) |
| 71 | \( 1 - 4.55T + 71T^{2} \) |
| 73 | \( 1 - 0.550T + 73T^{2} \) |
| 79 | \( 1 + 14.7T + 79T^{2} \) |
| 83 | \( 1 + 1.31T + 83T^{2} \) |
| 89 | \( 1 + 2.92T + 89T^{2} \) |
| 97 | \( 1 - 7.05T + 97T^{2} \) |
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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
−9.713707927343915216090958978413, −9.375367327295161097791720772940, −8.439909907875675438733298105496, −7.35042300481476939401201421166, −6.10535298640512714976655957394, −5.66418975365557481922508188083, −4.90562599962991481274150885673, −3.68469447828171663047595267811, −2.90548166630611780393694547630, −1.20723396337655314864028270521,
1.20723396337655314864028270521, 2.90548166630611780393694547630, 3.68469447828171663047595267811, 4.90562599962991481274150885673, 5.66418975365557481922508188083, 6.10535298640512714976655957394, 7.35042300481476939401201421166, 8.439909907875675438733298105496, 9.375367327295161097791720772940, 9.713707927343915216090958978413
