| L(s) = 1 | − 2-s + 3-s + 4-s + 2.56·5-s − 6-s + 2.56·7-s − 8-s + 9-s − 2.56·10-s + 5.12·11-s + 12-s + 3.12·13-s − 2.56·14-s + 2.56·15-s + 16-s − 2.56·17-s − 18-s − 6.56·19-s + 2.56·20-s + 2.56·21-s − 5.12·22-s − 23-s − 24-s + 1.56·25-s − 3.12·26-s + 27-s + 2.56·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s + 1.14·5-s − 0.408·6-s + 0.968·7-s − 0.353·8-s + 0.333·9-s − 0.810·10-s + 1.54·11-s + 0.288·12-s + 0.866·13-s − 0.684·14-s + 0.661·15-s + 0.250·16-s − 0.621·17-s − 0.235·18-s − 1.50·19-s + 0.572·20-s + 0.558·21-s − 1.09·22-s − 0.208·23-s − 0.204·24-s + 0.312·25-s − 0.612·26-s + 0.192·27-s + 0.484·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4002 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4002 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.797066147\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.797066147\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 + T \) |
| good | 5 | \( 1 - 2.56T + 5T^{2} \) |
| 7 | \( 1 - 2.56T + 7T^{2} \) |
| 11 | \( 1 - 5.12T + 11T^{2} \) |
| 13 | \( 1 - 3.12T + 13T^{2} \) |
| 17 | \( 1 + 2.56T + 17T^{2} \) |
| 19 | \( 1 + 6.56T + 19T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 8.56T + 37T^{2} \) |
| 41 | \( 1 - 8.56T + 41T^{2} \) |
| 43 | \( 1 - 6.56T + 43T^{2} \) |
| 47 | \( 1 + 3.68T + 47T^{2} \) |
| 53 | \( 1 + 1.12T + 53T^{2} \) |
| 59 | \( 1 - 3.68T + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 - 7.12T + 67T^{2} \) |
| 71 | \( 1 - 8T + 71T^{2} \) |
| 73 | \( 1 + 12.2T + 73T^{2} \) |
| 79 | \( 1 + 4.24T + 79T^{2} \) |
| 83 | \( 1 + 3.12T + 83T^{2} \) |
| 89 | \( 1 + 7.36T + 89T^{2} \) |
| 97 | \( 1 + 8T + 97T^{2} \) |
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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
−8.527123084686993755576046113273, −8.008711947197061390407654179175, −6.97687652327412040303079443253, −6.28519106395986755027539087191, −5.82065446891821920264987102351, −4.47554255424077955242147097606, −3.92348113563401435787925842836, −2.54026772422474172275614036592, −1.84737390275365766567595932958, −1.16555889038917173088536466973,
1.16555889038917173088536466973, 1.84737390275365766567595932958, 2.54026772422474172275614036592, 3.92348113563401435787925842836, 4.47554255424077955242147097606, 5.82065446891821920264987102351, 6.28519106395986755027539087191, 6.97687652327412040303079443253, 8.008711947197061390407654179175, 8.527123084686993755576046113273
