| L(s) = 1 | − 0.945·3-s − 3.79·5-s − 1.93·7-s − 2.10·9-s − 5.03·11-s − 5.05·13-s + 3.58·15-s + 3.40·17-s − 2.70·19-s + 1.83·21-s + 7.23·23-s + 9.37·25-s + 4.82·27-s + 3.66·29-s − 2.52·31-s + 4.76·33-s + 7.35·35-s − 1.96·37-s + 4.77·39-s − 4.85·41-s + 4.55·43-s + 7.98·45-s − 3.24·47-s − 3.24·49-s − 3.21·51-s − 12.8·53-s + 19.1·55-s + ⋯ |
| L(s) = 1 | − 0.545·3-s − 1.69·5-s − 0.732·7-s − 0.701·9-s − 1.51·11-s − 1.40·13-s + 0.925·15-s + 0.825·17-s − 0.621·19-s + 0.400·21-s + 1.50·23-s + 1.87·25-s + 0.929·27-s + 0.680·29-s − 0.453·31-s + 0.829·33-s + 1.24·35-s − 0.323·37-s + 0.764·39-s − 0.757·41-s + 0.693·43-s + 1.19·45-s − 0.473·47-s − 0.462·49-s − 0.450·51-s − 1.76·53-s + 2.57·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2587433717\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2587433717\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 167 | \( 1 - T \) |
| good | 3 | \( 1 + 0.945T + 3T^{2} \) |
| 5 | \( 1 + 3.79T + 5T^{2} \) |
| 7 | \( 1 + 1.93T + 7T^{2} \) |
| 11 | \( 1 + 5.03T + 11T^{2} \) |
| 13 | \( 1 + 5.05T + 13T^{2} \) |
| 17 | \( 1 - 3.40T + 17T^{2} \) |
| 19 | \( 1 + 2.70T + 19T^{2} \) |
| 23 | \( 1 - 7.23T + 23T^{2} \) |
| 29 | \( 1 - 3.66T + 29T^{2} \) |
| 31 | \( 1 + 2.52T + 31T^{2} \) |
| 37 | \( 1 + 1.96T + 37T^{2} \) |
| 41 | \( 1 + 4.85T + 41T^{2} \) |
| 43 | \( 1 - 4.55T + 43T^{2} \) |
| 47 | \( 1 + 3.24T + 47T^{2} \) |
| 53 | \( 1 + 12.8T + 53T^{2} \) |
| 59 | \( 1 + 5.99T + 59T^{2} \) |
| 61 | \( 1 + 8.02T + 61T^{2} \) |
| 67 | \( 1 - 13.6T + 67T^{2} \) |
| 71 | \( 1 - 6.53T + 71T^{2} \) |
| 73 | \( 1 + 5.85T + 73T^{2} \) |
| 79 | \( 1 + 3.19T + 79T^{2} \) |
| 83 | \( 1 + 5.99T + 83T^{2} \) |
| 89 | \( 1 - 17.8T + 89T^{2} \) |
| 97 | \( 1 + 12.3T + 97T^{2} \) |
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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
−9.710299905435810479428914615321, −8.636789363748891855581832003244, −7.87135397217945060478358731250, −7.32316657218070829967438592213, −6.41632154614273132720439051702, −5.16389780343164351543256539938, −4.75962043316923232550623521364, −3.33128766523778013813548126465, −2.78727783506958649067300501946, −0.35277196741851137542048016380,
0.35277196741851137542048016380, 2.78727783506958649067300501946, 3.33128766523778013813548126465, 4.75962043316923232550623521364, 5.16389780343164351543256539938, 6.41632154614273132720439051702, 7.32316657218070829967438592213, 7.87135397217945060478358731250, 8.636789363748891855581832003244, 9.710299905435810479428914615321
