| L(s) = 1 | − 3-s − 2.11·5-s − 0.802·7-s + 9-s − 1.53·11-s + 6.15·13-s + 2.11·15-s + 2.02·17-s − 8.09·19-s + 0.802·21-s − 1.69·23-s − 0.508·25-s − 27-s + 8.39·29-s + 1.83·31-s + 1.53·33-s + 1.70·35-s − 9.93·37-s − 6.15·39-s − 2.89·41-s + 2.94·43-s − 2.11·45-s − 7.81·47-s − 6.35·49-s − 2.02·51-s + 9.25·53-s + 3.24·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.947·5-s − 0.303·7-s + 0.333·9-s − 0.461·11-s + 1.70·13-s + 0.547·15-s + 0.491·17-s − 1.85·19-s + 0.175·21-s − 0.353·23-s − 0.101·25-s − 0.192·27-s + 1.55·29-s + 0.329·31-s + 0.266·33-s + 0.287·35-s − 1.63·37-s − 0.985·39-s − 0.451·41-s + 0.449·43-s − 0.315·45-s − 1.13·47-s − 0.907·49-s − 0.283·51-s + 1.27·53-s + 0.437·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9281827295\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9281827295\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 167 | \( 1 - T \) |
| good | 5 | \( 1 + 2.11T + 5T^{2} \) |
| 7 | \( 1 + 0.802T + 7T^{2} \) |
| 11 | \( 1 + 1.53T + 11T^{2} \) |
| 13 | \( 1 - 6.15T + 13T^{2} \) |
| 17 | \( 1 - 2.02T + 17T^{2} \) |
| 19 | \( 1 + 8.09T + 19T^{2} \) |
| 23 | \( 1 + 1.69T + 23T^{2} \) |
| 29 | \( 1 - 8.39T + 29T^{2} \) |
| 31 | \( 1 - 1.83T + 31T^{2} \) |
| 37 | \( 1 + 9.93T + 37T^{2} \) |
| 41 | \( 1 + 2.89T + 41T^{2} \) |
| 43 | \( 1 - 2.94T + 43T^{2} \) |
| 47 | \( 1 + 7.81T + 47T^{2} \) |
| 53 | \( 1 - 9.25T + 53T^{2} \) |
| 59 | \( 1 - 5.76T + 59T^{2} \) |
| 61 | \( 1 + 0.0936T + 61T^{2} \) |
| 67 | \( 1 + 8.66T + 67T^{2} \) |
| 71 | \( 1 + 8.00T + 71T^{2} \) |
| 73 | \( 1 - 1.52T + 73T^{2} \) |
| 79 | \( 1 - 15.5T + 79T^{2} \) |
| 83 | \( 1 - 13.8T + 83T^{2} \) |
| 89 | \( 1 - 11.9T + 89T^{2} \) |
| 97 | \( 1 + 13.7T + 97T^{2} \) |
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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
−8.325052689956014775296882548544, −7.907897474619542483718806706795, −6.73096368225118402984585566004, −6.39920594135327112634349738336, −5.53252006831522634204719319257, −4.57557622254836902566752646990, −3.89143310315988424453487288031, −3.18930661875265285517534679125, −1.83829601557266822708867650277, −0.56510050326373827071438578856,
0.56510050326373827071438578856, 1.83829601557266822708867650277, 3.18930661875265285517534679125, 3.89143310315988424453487288031, 4.57557622254836902566752646990, 5.53252006831522634204719319257, 6.39920594135327112634349738336, 6.73096368225118402984585566004, 7.907897474619542483718806706795, 8.325052689956014775296882548544
