| L(s) = 1 | − 0.167·2-s − 3-s − 1.97·4-s − 1.16·5-s + 0.167·6-s + 0.665·8-s + 9-s + 0.195·10-s − 1.80·11-s + 1.97·12-s − 6.97·13-s + 1.16·15-s + 3.83·16-s − 6.13·17-s − 0.167·18-s − 1.80·19-s + 2.30·20-s + 0.302·22-s − 23-s − 0.665·24-s − 3.63·25-s + 1.16·26-s − 27-s + 5.46·29-s − 0.195·30-s + 2.19·31-s − 1.97·32-s + ⋯ |
| L(s) = 1 | − 0.118·2-s − 0.577·3-s − 0.985·4-s − 0.522·5-s + 0.0683·6-s + 0.235·8-s + 0.333·9-s + 0.0618·10-s − 0.544·11-s + 0.569·12-s − 1.93·13-s + 0.301·15-s + 0.958·16-s − 1.48·17-s − 0.0394·18-s − 0.413·19-s + 0.514·20-s + 0.0644·22-s − 0.208·23-s − 0.135·24-s − 0.727·25-s + 0.228·26-s − 0.192·27-s + 1.01·29-s − 0.0356·30-s + 0.394·31-s − 0.348·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1804858434\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1804858434\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 7 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 2 | \( 1 + 0.167T + 2T^{2} \) |
| 5 | \( 1 + 1.16T + 5T^{2} \) |
| 11 | \( 1 + 1.80T + 11T^{2} \) |
| 13 | \( 1 + 6.97T + 13T^{2} \) |
| 17 | \( 1 + 6.13T + 17T^{2} \) |
| 19 | \( 1 + 1.80T + 19T^{2} \) |
| 29 | \( 1 - 5.46T + 29T^{2} \) |
| 31 | \( 1 - 2.19T + 31T^{2} \) |
| 37 | \( 1 + 1.46T + 37T^{2} \) |
| 41 | \( 1 + 9.74T + 41T^{2} \) |
| 43 | \( 1 + 6.63T + 43T^{2} \) |
| 47 | \( 1 - 1.94T + 47T^{2} \) |
| 53 | \( 1 - 1.16T + 53T^{2} \) |
| 59 | \( 1 + 9.11T + 59T^{2} \) |
| 61 | \( 1 - 4.63T + 61T^{2} \) |
| 67 | \( 1 + 15.4T + 67T^{2} \) |
| 71 | \( 1 + 9.25T + 71T^{2} \) |
| 73 | \( 1 - 0.474T + 73T^{2} \) |
| 79 | \( 1 - 7.46T + 79T^{2} \) |
| 83 | \( 1 + 8.13T + 83T^{2} \) |
| 89 | \( 1 + 1.02T + 89T^{2} \) |
| 97 | \( 1 - 8.47T + 97T^{2} \) |
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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
−8.597179403530644557845992948883, −7.894328232323275159957424969140, −7.20663605520152693178792292834, −6.41138665147779275924755758272, −5.36270397000094329740400296419, −4.66245048605440427534088636445, −4.34153736690433087421148271903, −3.08399025486773192999829271992, −1.95685428237061879761503200943, −0.25326714943112172465414344513,
0.25326714943112172465414344513, 1.95685428237061879761503200943, 3.08399025486773192999829271992, 4.34153736690433087421148271903, 4.66245048605440427534088636445, 5.36270397000094329740400296419, 6.41138665147779275924755758272, 7.20663605520152693178792292834, 7.894328232323275159957424969140, 8.597179403530644557845992948883
