| L(s) = 1 | − 1.89·2-s − 3-s + 1.58·4-s + 1.89·6-s − 1.92·7-s + 0.788·8-s + 9-s − 2.02·11-s − 1.58·12-s − 3.44·13-s + 3.64·14-s − 4.65·16-s − 7.60·17-s − 1.89·18-s + 0.268·19-s + 1.92·21-s + 3.83·22-s − 0.714·23-s − 0.788·24-s + 6.51·26-s − 27-s − 3.04·28-s + 5.15·29-s − 31-s + 7.24·32-s + 2.02·33-s + 14.3·34-s + ⋯ |
| L(s) = 1 | − 1.33·2-s − 0.577·3-s + 0.791·4-s + 0.772·6-s − 0.727·7-s + 0.278·8-s + 0.333·9-s − 0.610·11-s − 0.457·12-s − 0.954·13-s + 0.973·14-s − 1.16·16-s − 1.84·17-s − 0.446·18-s + 0.0615·19-s + 0.420·21-s + 0.816·22-s − 0.149·23-s − 0.161·24-s + 1.27·26-s − 0.192·27-s − 0.575·28-s + 0.956·29-s − 0.179·31-s + 1.28·32-s + 0.352·33-s + 2.46·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2325 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2391501759\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2391501759\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 31 | \( 1 + T \) |
| good | 2 | \( 1 + 1.89T + 2T^{2} \) |
| 7 | \( 1 + 1.92T + 7T^{2} \) |
| 11 | \( 1 + 2.02T + 11T^{2} \) |
| 13 | \( 1 + 3.44T + 13T^{2} \) |
| 17 | \( 1 + 7.60T + 17T^{2} \) |
| 19 | \( 1 - 0.268T + 19T^{2} \) |
| 23 | \( 1 + 0.714T + 23T^{2} \) |
| 29 | \( 1 - 5.15T + 29T^{2} \) |
| 37 | \( 1 - 1.83T + 37T^{2} \) |
| 41 | \( 1 - 1.23T + 41T^{2} \) |
| 43 | \( 1 + 7.70T + 43T^{2} \) |
| 47 | \( 1 - 3.33T + 47T^{2} \) |
| 53 | \( 1 + 3.17T + 53T^{2} \) |
| 59 | \( 1 - 7.01T + 59T^{2} \) |
| 61 | \( 1 + 11.7T + 61T^{2} \) |
| 67 | \( 1 + 11.3T + 67T^{2} \) |
| 71 | \( 1 + 12.9T + 71T^{2} \) |
| 73 | \( 1 + 0.570T + 73T^{2} \) |
| 79 | \( 1 - 14.9T + 79T^{2} \) |
| 83 | \( 1 - 13.6T + 83T^{2} \) |
| 89 | \( 1 + 1.63T + 89T^{2} \) |
| 97 | \( 1 + 3.13T + 97T^{2} \) |
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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
−9.124781231668713667666100457919, −8.329799217399838265281669396572, −7.51697893907113865433846205410, −6.83099060944010951395601976505, −6.21593803548796856879589248734, −4.97727317326935130072449551065, −4.35896963569583172381556432150, −2.85464198186049749921287183582, −1.88903930320480519003066038234, −0.38025538155338134031545431220,
0.38025538155338134031545431220, 1.88903930320480519003066038234, 2.85464198186049749921287183582, 4.35896963569583172381556432150, 4.97727317326935130072449551065, 6.21593803548796856879589248734, 6.83099060944010951395601976505, 7.51697893907113865433846205410, 8.329799217399838265281669396572, 9.124781231668713667666100457919
