| L(s) = 1 | − 2-s − 3-s + 4-s + 3.32·5-s + 6-s + 5.20·7-s − 8-s + 9-s − 3.32·10-s + 3.32·11-s − 12-s − 1.32·13-s − 5.20·14-s − 3.32·15-s + 16-s + 4·17-s − 18-s − 1.20·19-s + 3.32·20-s − 5.20·21-s − 3.32·22-s + 23-s + 24-s + 6.07·25-s + 1.32·26-s − 27-s + 5.20·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s + 1.48·5-s + 0.408·6-s + 1.96·7-s − 0.353·8-s + 0.333·9-s − 1.05·10-s + 1.00·11-s − 0.288·12-s − 0.368·13-s − 1.39·14-s − 0.859·15-s + 0.250·16-s + 0.970·17-s − 0.235·18-s − 0.275·19-s + 0.744·20-s − 1.13·21-s − 0.709·22-s + 0.208·23-s + 0.204·24-s + 1.21·25-s + 0.260·26-s − 0.192·27-s + 0.983·28-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
| \(L(1)\) |
\(\approx\) |
\(2.244021788\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.244021788\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 3.32T + 5T^{2} \) |
| 7 | \( 1 - 5.20T + 7T^{2} \) |
| 11 | \( 1 - 3.32T + 11T^{2} \) |
| 13 | \( 1 + 1.32T + 13T^{2} \) |
| 17 | \( 1 - 4T + 17T^{2} \) |
| 19 | \( 1 + 1.20T + 19T^{2} \) |
| 31 | \( 1 + 1.32T + 31T^{2} \) |
| 37 | \( 1 + 5.07T + 37T^{2} \) |
| 41 | \( 1 + 0.924T + 41T^{2} \) |
| 43 | \( 1 - 11.8T + 43T^{2} \) |
| 47 | \( 1 + 10.6T + 47T^{2} \) |
| 53 | \( 1 - 6.65T + 53T^{2} \) |
| 59 | \( 1 - 11.3T + 59T^{2} \) |
| 61 | \( 1 - 1.32T + 61T^{2} \) |
| 67 | \( 1 - 1.32T + 67T^{2} \) |
| 71 | \( 1 + 9.58T + 71T^{2} \) |
| 73 | \( 1 + 2.25T + 73T^{2} \) |
| 79 | \( 1 + 8.40T + 79T^{2} \) |
| 83 | \( 1 + 9.60T + 83T^{2} \) |
| 89 | \( 1 + 10.6T + 89T^{2} \) |
| 97 | \( 1 - 13.4T + 97T^{2} \) |
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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
−8.647132380316788731141949980551, −7.68100193527988351393627877972, −7.08479577864685174762188385256, −6.17549682787975760287567984057, −5.51089934227139874907993617740, −4.99473806584577512653746352047, −3.98432218213623488561902477217, −2.46490656574028925716209401381, −1.64046058496129349032308733341, −1.14681366607036368655386826491,
1.14681366607036368655386826491, 1.64046058496129349032308733341, 2.46490656574028925716209401381, 3.98432218213623488561902477217, 4.99473806584577512653746352047, 5.51089934227139874907993617740, 6.17549682787975760287567984057, 7.08479577864685174762188385256, 7.68100193527988351393627877972, 8.647132380316788731141949980551
