| L(s) = 1 | + 2-s + 2.35·3-s + 4-s + 3.31·5-s + 2.35·6-s + 8-s + 2.53·9-s + 3.31·10-s − 0.960·11-s + 2.35·12-s + 1.18·13-s + 7.79·15-s + 16-s − 6.98·17-s + 2.53·18-s + 4.81·19-s + 3.31·20-s − 0.960·22-s + 5.44·23-s + 2.35·24-s + 5.98·25-s + 1.18·26-s − 1.09·27-s + 0.497·29-s + 7.79·30-s + 0.463·31-s + 32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.35·3-s + 0.5·4-s + 1.48·5-s + 0.960·6-s + 0.353·8-s + 0.845·9-s + 1.04·10-s − 0.289·11-s + 0.679·12-s + 0.328·13-s + 2.01·15-s + 0.250·16-s − 1.69·17-s + 0.597·18-s + 1.10·19-s + 0.740·20-s − 0.204·22-s + 1.13·23-s + 0.480·24-s + 1.19·25-s + 0.232·26-s − 0.209·27-s + 0.0923·29-s + 1.42·30-s + 0.0832·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.481558427\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.481558427\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 7 | \( 1 \) |
| 41 | \( 1 - T \) |
| good | 3 | \( 1 - 2.35T + 3T^{2} \) |
| 5 | \( 1 - 3.31T + 5T^{2} \) |
| 11 | \( 1 + 0.960T + 11T^{2} \) |
| 13 | \( 1 - 1.18T + 13T^{2} \) |
| 17 | \( 1 + 6.98T + 17T^{2} \) |
| 19 | \( 1 - 4.81T + 19T^{2} \) |
| 23 | \( 1 - 5.44T + 23T^{2} \) |
| 29 | \( 1 - 0.497T + 29T^{2} \) |
| 31 | \( 1 - 0.463T + 31T^{2} \) |
| 37 | \( 1 + 6.70T + 37T^{2} \) |
| 43 | \( 1 - 5.09T + 43T^{2} \) |
| 47 | \( 1 - 2.73T + 47T^{2} \) |
| 53 | \( 1 + 0.418T + 53T^{2} \) |
| 59 | \( 1 + 11.1T + 59T^{2} \) |
| 61 | \( 1 + 3.94T + 61T^{2} \) |
| 67 | \( 1 - 8.37T + 67T^{2} \) |
| 71 | \( 1 - 0.830T + 71T^{2} \) |
| 73 | \( 1 + 1.63T + 73T^{2} \) |
| 79 | \( 1 + 4.86T + 79T^{2} \) |
| 83 | \( 1 - 10.4T + 83T^{2} \) |
| 89 | \( 1 - 8.46T + 89T^{2} \) |
| 97 | \( 1 + 8.72T + 97T^{2} \) |
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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
−8.617140899880853321348040583337, −7.66268458520146628368406764520, −6.92034875642020222417833907735, −6.22015367266251868502846743984, −5.39643727433081906464103966343, −4.70305986427575127719742719172, −3.65844875383600319975728593394, −2.81792586554378225579848736896, −2.29614750774010149438880631057, −1.43580468721486284283281185859,
1.43580468721486284283281185859, 2.29614750774010149438880631057, 2.81792586554378225579848736896, 3.65844875383600319975728593394, 4.70305986427575127719742719172, 5.39643727433081906464103966343, 6.22015367266251868502846743984, 6.92034875642020222417833907735, 7.66268458520146628368406764520, 8.617140899880853321348040583337
