| L(s) = 1 | − 2.26·2-s − 1.83·3-s + 3.14·4-s − 3.37·5-s + 4.15·6-s + 3.93·7-s − 2.59·8-s + 0.354·9-s + 7.64·10-s − 2.11·11-s − 5.76·12-s − 5.61·13-s − 8.92·14-s + 6.17·15-s − 0.399·16-s − 5.47·17-s − 0.804·18-s − 1.80·19-s − 10.6·20-s − 7.20·21-s + 4.78·22-s + 7.43·23-s + 4.75·24-s + 6.36·25-s + 12.7·26-s + 4.84·27-s + 12.3·28-s + ⋯ |
| L(s) = 1 | − 1.60·2-s − 1.05·3-s + 1.57·4-s − 1.50·5-s + 1.69·6-s + 1.48·7-s − 0.918·8-s + 0.118·9-s + 2.41·10-s − 0.636·11-s − 1.66·12-s − 1.55·13-s − 2.38·14-s + 1.59·15-s − 0.0998·16-s − 1.32·17-s − 0.189·18-s − 0.415·19-s − 2.37·20-s − 1.57·21-s + 1.02·22-s + 1.55·23-s + 0.970·24-s + 1.27·25-s + 2.49·26-s + 0.932·27-s + 2.33·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 503 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 503 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2198787393\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2198787393\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 503 | \( 1 - T \) |
| good | 2 | \( 1 + 2.26T + 2T^{2} \) |
| 3 | \( 1 + 1.83T + 3T^{2} \) |
| 5 | \( 1 + 3.37T + 5T^{2} \) |
| 7 | \( 1 - 3.93T + 7T^{2} \) |
| 11 | \( 1 + 2.11T + 11T^{2} \) |
| 13 | \( 1 + 5.61T + 13T^{2} \) |
| 17 | \( 1 + 5.47T + 17T^{2} \) |
| 19 | \( 1 + 1.80T + 19T^{2} \) |
| 23 | \( 1 - 7.43T + 23T^{2} \) |
| 29 | \( 1 + 3.93T + 29T^{2} \) |
| 31 | \( 1 + 0.478T + 31T^{2} \) |
| 37 | \( 1 - 9.53T + 37T^{2} \) |
| 41 | \( 1 + 11.7T + 41T^{2} \) |
| 43 | \( 1 + 3.65T + 43T^{2} \) |
| 47 | \( 1 - 6.91T + 47T^{2} \) |
| 53 | \( 1 - 9.78T + 53T^{2} \) |
| 59 | \( 1 - 11.8T + 59T^{2} \) |
| 61 | \( 1 - 10.2T + 61T^{2} \) |
| 67 | \( 1 + 7.56T + 67T^{2} \) |
| 71 | \( 1 + 0.790T + 71T^{2} \) |
| 73 | \( 1 + 5.39T + 73T^{2} \) |
| 79 | \( 1 - 9.34T + 79T^{2} \) |
| 83 | \( 1 + 2.16T + 83T^{2} \) |
| 89 | \( 1 - 5.51T + 89T^{2} \) |
| 97 | \( 1 - 15.8T + 97T^{2} \) |
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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
−10.98197892197834773430444174597, −10.28649447030170667832854448760, −8.886190685702138760295822690723, −8.274272552296845060608125335997, −7.43913611211707965563808085252, −6.92264545723134351152739059023, −5.14597072517186142302811516011, −4.50756906352755635589768245344, −2.35061665825409653532692485765, −0.54552096623706041305675961124,
0.54552096623706041305675961124, 2.35061665825409653532692485765, 4.50756906352755635589768245344, 5.14597072517186142302811516011, 6.92264545723134351152739059023, 7.43913611211707965563808085252, 8.274272552296845060608125335997, 8.886190685702138760295822690723, 10.28649447030170667832854448760, 10.98197892197834773430444174597
