| L(s) = 1 | + 2.50·2-s + 3-s + 4.27·4-s + 4.39·5-s + 2.50·6-s + 7-s + 5.69·8-s + 9-s + 11.0·10-s − 4.36·11-s + 4.27·12-s + 1.96·13-s + 2.50·14-s + 4.39·15-s + 5.72·16-s − 7.50·17-s + 2.50·18-s + 3.29·19-s + 18.8·20-s + 21-s − 10.9·22-s − 9.03·23-s + 5.69·24-s + 14.3·25-s + 4.93·26-s + 27-s + 4.27·28-s + ⋯ |
| L(s) = 1 | + 1.77·2-s + 0.577·3-s + 2.13·4-s + 1.96·5-s + 1.02·6-s + 0.377·7-s + 2.01·8-s + 0.333·9-s + 3.48·10-s − 1.31·11-s + 1.23·12-s + 0.546·13-s + 0.669·14-s + 1.13·15-s + 1.43·16-s − 1.81·17-s + 0.590·18-s + 0.754·19-s + 4.20·20-s + 0.218·21-s − 2.33·22-s − 1.88·23-s + 1.16·24-s + 2.86·25-s + 0.967·26-s + 0.192·27-s + 0.807·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4011 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4011 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(9.675733164\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.675733164\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 191 | \( 1 - T \) |
| good | 2 | \( 1 - 2.50T + 2T^{2} \) |
| 5 | \( 1 - 4.39T + 5T^{2} \) |
| 11 | \( 1 + 4.36T + 11T^{2} \) |
| 13 | \( 1 - 1.96T + 13T^{2} \) |
| 17 | \( 1 + 7.50T + 17T^{2} \) |
| 19 | \( 1 - 3.29T + 19T^{2} \) |
| 23 | \( 1 + 9.03T + 23T^{2} \) |
| 29 | \( 1 - 6.18T + 29T^{2} \) |
| 31 | \( 1 + 9.78T + 31T^{2} \) |
| 37 | \( 1 + 0.428T + 37T^{2} \) |
| 41 | \( 1 + 1.19T + 41T^{2} \) |
| 43 | \( 1 - 5.86T + 43T^{2} \) |
| 47 | \( 1 - 3.06T + 47T^{2} \) |
| 53 | \( 1 - 11.2T + 53T^{2} \) |
| 59 | \( 1 - 0.735T + 59T^{2} \) |
| 61 | \( 1 + 2.69T + 61T^{2} \) |
| 67 | \( 1 + 14.3T + 67T^{2} \) |
| 71 | \( 1 - 12.8T + 71T^{2} \) |
| 73 | \( 1 - 3.96T + 73T^{2} \) |
| 79 | \( 1 + 12.4T + 79T^{2} \) |
| 83 | \( 1 - 1.24T + 83T^{2} \) |
| 89 | \( 1 + 6.13T + 89T^{2} \) |
| 97 | \( 1 + 17.0T + 97T^{2} \) |
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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
−8.453888090968261398754045893249, −7.38611122676835382048837519871, −6.67698538052637706594733030474, −5.88341321052662161682279737017, −5.52133572742578124550038915357, −4.75249750311790335523344281995, −3.97720543120328867904370269507, −2.80372713281352169331459723266, −2.31146992839777907420927349516, −1.71096609619807990770948103696,
1.71096609619807990770948103696, 2.31146992839777907420927349516, 2.80372713281352169331459723266, 3.97720543120328867904370269507, 4.75249750311790335523344281995, 5.52133572742578124550038915357, 5.88341321052662161682279737017, 6.67698538052637706594733030474, 7.38611122676835382048837519871, 8.453888090968261398754045893249
