| L(s) = 1 | + 2-s − 3-s + 4-s + 2.25·5-s − 6-s + 3.73·7-s + 8-s + 9-s + 2.25·10-s + 3.84·11-s − 12-s − 4.08·13-s + 3.73·14-s − 2.25·15-s + 16-s − 17-s + 18-s − 1.16·19-s + 2.25·20-s − 3.73·21-s + 3.84·22-s + 5.03·23-s − 24-s + 0.105·25-s − 4.08·26-s − 27-s + 3.73·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s + 1.01·5-s − 0.408·6-s + 1.41·7-s + 0.353·8-s + 0.333·9-s + 0.714·10-s + 1.15·11-s − 0.288·12-s − 1.13·13-s + 0.998·14-s − 0.583·15-s + 0.250·16-s − 0.242·17-s + 0.235·18-s − 0.267·19-s + 0.505·20-s − 0.815·21-s + 0.818·22-s + 1.05·23-s − 0.204·24-s + 0.0210·25-s − 0.800·26-s − 0.192·27-s + 0.705·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.199052594\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.199052594\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| 59 | \( 1 + T \) |
| good | 5 | \( 1 - 2.25T + 5T^{2} \) |
| 7 | \( 1 - 3.73T + 7T^{2} \) |
| 11 | \( 1 - 3.84T + 11T^{2} \) |
| 13 | \( 1 + 4.08T + 13T^{2} \) |
| 19 | \( 1 + 1.16T + 19T^{2} \) |
| 23 | \( 1 - 5.03T + 23T^{2} \) |
| 29 | \( 1 - 6.70T + 29T^{2} \) |
| 31 | \( 1 + 7.12T + 31T^{2} \) |
| 37 | \( 1 + 2.75T + 37T^{2} \) |
| 41 | \( 1 - 8.36T + 41T^{2} \) |
| 43 | \( 1 - 10.5T + 43T^{2} \) |
| 47 | \( 1 - 0.0503T + 47T^{2} \) |
| 53 | \( 1 + 0.972T + 53T^{2} \) |
| 61 | \( 1 + 6.11T + 61T^{2} \) |
| 67 | \( 1 + 0.356T + 67T^{2} \) |
| 71 | \( 1 - 5.02T + 71T^{2} \) |
| 73 | \( 1 + 5.24T + 73T^{2} \) |
| 79 | \( 1 - 0.531T + 79T^{2} \) |
| 83 | \( 1 - 5.67T + 83T^{2} \) |
| 89 | \( 1 - 13.4T + 89T^{2} \) |
| 97 | \( 1 + 7.52T + 97T^{2} \) |
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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
−7.81647655010576531936076441942, −7.23654768545215729451539466715, −6.48385348657420838394968199905, −5.85430642930383176490000517916, −5.11021896353053682170482052335, −4.67498972480100942030278600403, −3.92612333342897831938874748313, −2.61241176035099434387262271426, −1.88785560161936967126275458425, −1.08863142844628626707575259056,
1.08863142844628626707575259056, 1.88785560161936967126275458425, 2.61241176035099434387262271426, 3.92612333342897831938874748313, 4.67498972480100942030278600403, 5.11021896353053682170482052335, 5.85430642930383176490000517916, 6.48385348657420838394968199905, 7.23654768545215729451539466715, 7.81647655010576531936076441942
