| L(s) = 1 | − 0.870·2-s − 1.24·4-s + 3.50·5-s − 4.57·7-s + 2.82·8-s − 3.04·10-s − 11-s + 5.30·13-s + 3.97·14-s + 0.0307·16-s − 2.88·17-s − 3.82·19-s − 4.35·20-s + 0.870·22-s − 7.78·23-s + 7.26·25-s − 4.61·26-s + 5.68·28-s + 0.514·29-s − 5.54·31-s − 5.67·32-s + 2.51·34-s − 16.0·35-s + 0.537·37-s + 3.32·38-s + 9.88·40-s − 3.66·41-s + ⋯ |
| L(s) = 1 | − 0.615·2-s − 0.621·4-s + 1.56·5-s − 1.72·7-s + 0.997·8-s − 0.963·10-s − 0.301·11-s + 1.47·13-s + 1.06·14-s + 0.00769·16-s − 0.699·17-s − 0.877·19-s − 0.973·20-s + 0.185·22-s − 1.62·23-s + 1.45·25-s − 0.904·26-s + 1.07·28-s + 0.0954·29-s − 0.996·31-s − 1.00·32-s + 0.430·34-s − 2.70·35-s + 0.0883·37-s + 0.540·38-s + 1.56·40-s − 0.572·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.033627092\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.033627092\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 + T \) |
| 61 | \( 1 - T \) |
| good | 2 | \( 1 + 0.870T + 2T^{2} \) |
| 5 | \( 1 - 3.50T + 5T^{2} \) |
| 7 | \( 1 + 4.57T + 7T^{2} \) |
| 13 | \( 1 - 5.30T + 13T^{2} \) |
| 17 | \( 1 + 2.88T + 17T^{2} \) |
| 19 | \( 1 + 3.82T + 19T^{2} \) |
| 23 | \( 1 + 7.78T + 23T^{2} \) |
| 29 | \( 1 - 0.514T + 29T^{2} \) |
| 31 | \( 1 + 5.54T + 31T^{2} \) |
| 37 | \( 1 - 0.537T + 37T^{2} \) |
| 41 | \( 1 + 3.66T + 41T^{2} \) |
| 43 | \( 1 - 11.1T + 43T^{2} \) |
| 47 | \( 1 - 12.3T + 47T^{2} \) |
| 53 | \( 1 - 9.47T + 53T^{2} \) |
| 59 | \( 1 - 1.31T + 59T^{2} \) |
| 67 | \( 1 + 12.7T + 67T^{2} \) |
| 71 | \( 1 + 5.58T + 71T^{2} \) |
| 73 | \( 1 - 8.67T + 73T^{2} \) |
| 79 | \( 1 - 9.71T + 79T^{2} \) |
| 83 | \( 1 + 10.8T + 83T^{2} \) |
| 89 | \( 1 - 10.0T + 89T^{2} \) |
| 97 | \( 1 + 16.0T + 97T^{2} \) |
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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
−8.399704493814727744206935909564, −7.32500345922177546225033190013, −6.53380608960149732386263915055, −5.89545804302697509423532286333, −5.64123559494066461838830196095, −4.25806187811966978277481475995, −3.72511402550389503636443082811, −2.56593539204211682010809950193, −1.79689777650752037769613093266, −0.58263028731160210491392830420,
0.58263028731160210491392830420, 1.79689777650752037769613093266, 2.56593539204211682010809950193, 3.72511402550389503636443082811, 4.25806187811966978277481475995, 5.64123559494066461838830196095, 5.89545804302697509423532286333, 6.53380608960149732386263915055, 7.32500345922177546225033190013, 8.399704493814727744206935909564
