| L(s) = 1 | + 2.53·2-s + 3-s + 4.42·4-s + 4.31·5-s + 2.53·6-s − 3.18·7-s + 6.13·8-s + 9-s + 10.9·10-s + 2.64·11-s + 4.42·12-s − 13-s − 8.07·14-s + 4.31·15-s + 6.69·16-s − 4.67·17-s + 2.53·18-s − 3.29·19-s + 19.0·20-s − 3.18·21-s + 6.71·22-s + 5.22·23-s + 6.13·24-s + 13.6·25-s − 2.53·26-s + 27-s − 14.0·28-s + ⋯ |
| L(s) = 1 | + 1.79·2-s + 0.577·3-s + 2.21·4-s + 1.93·5-s + 1.03·6-s − 1.20·7-s + 2.16·8-s + 0.333·9-s + 3.45·10-s + 0.798·11-s + 1.27·12-s − 0.277·13-s − 2.15·14-s + 1.11·15-s + 1.67·16-s − 1.13·17-s + 0.597·18-s − 0.755·19-s + 4.26·20-s − 0.695·21-s + 1.43·22-s + 1.08·23-s + 1.25·24-s + 2.72·25-s − 0.496·26-s + 0.192·27-s − 2.66·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(9.225239470\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.225239470\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| 103 | \( 1 + T \) |
| good | 2 | \( 1 - 2.53T + 2T^{2} \) |
| 5 | \( 1 - 4.31T + 5T^{2} \) |
| 7 | \( 1 + 3.18T + 7T^{2} \) |
| 11 | \( 1 - 2.64T + 11T^{2} \) |
| 17 | \( 1 + 4.67T + 17T^{2} \) |
| 19 | \( 1 + 3.29T + 19T^{2} \) |
| 23 | \( 1 - 5.22T + 23T^{2} \) |
| 29 | \( 1 + 3.48T + 29T^{2} \) |
| 31 | \( 1 - 1.50T + 31T^{2} \) |
| 37 | \( 1 - 0.214T + 37T^{2} \) |
| 41 | \( 1 + 0.0363T + 41T^{2} \) |
| 43 | \( 1 - 0.430T + 43T^{2} \) |
| 47 | \( 1 + 5.64T + 47T^{2} \) |
| 53 | \( 1 - 7.76T + 53T^{2} \) |
| 59 | \( 1 - 5.31T + 59T^{2} \) |
| 61 | \( 1 - 0.756T + 61T^{2} \) |
| 67 | \( 1 + 5.32T + 67T^{2} \) |
| 71 | \( 1 - 5.39T + 71T^{2} \) |
| 73 | \( 1 + 5.42T + 73T^{2} \) |
| 79 | \( 1 - 0.812T + 79T^{2} \) |
| 83 | \( 1 + 6.52T + 83T^{2} \) |
| 89 | \( 1 + 3.34T + 89T^{2} \) |
| 97 | \( 1 + 18.6T + 97T^{2} \) |
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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
−8.650202009578733904791406087679, −6.97997995044160494013394374809, −6.76311232753766863330222637627, −6.15496440747429300894020770821, −5.49912809476972989204809697607, −4.67499162488075956492061379904, −3.85775514641446797259682716021, −2.91017874597029757501346050492, −2.42658453233714317586983722626, −1.57730532691012521233603669165,
1.57730532691012521233603669165, 2.42658453233714317586983722626, 2.91017874597029757501346050492, 3.85775514641446797259682716021, 4.67499162488075956492061379904, 5.49912809476972989204809697607, 6.15496440747429300894020770821, 6.76311232753766863330222637627, 6.97997995044160494013394374809, 8.650202009578733904791406087679
