| L(s) = 1 | − 2-s + 3-s + 4-s − 5-s − 6-s − 8-s + 9-s + 10-s + 5·11-s + 12-s + 3·13-s − 15-s + 16-s + 4·17-s − 18-s + 4·19-s − 20-s − 5·22-s − 23-s − 24-s − 4·25-s − 3·26-s + 27-s − 29-s + 30-s + 5·31-s − 32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s − 0.353·8-s + 1/3·9-s + 0.316·10-s + 1.50·11-s + 0.288·12-s + 0.832·13-s − 0.258·15-s + 1/4·16-s + 0.970·17-s − 0.235·18-s + 0.917·19-s − 0.223·20-s − 1.06·22-s − 0.208·23-s − 0.204·24-s − 4/5·25-s − 0.588·26-s + 0.192·27-s − 0.185·29-s + 0.182·30-s + 0.898·31-s − 0.176·32-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
| \(L(1)\) |
\(\approx\) |
\(1.958439422\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.958439422\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 13 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 - 10 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 7 T + p T^{2} \) |
| 61 | \( 1 - T + p T^{2} \) |
| 67 | \( 1 + 13 T + p T^{2} \) |
| 71 | \( 1 + 5 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + 2 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
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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
−8.582903535103051671220586523393, −7.70107674785881678622186592031, −7.31715568476819843350858462741, −6.32891082747463643586427384833, −5.74985936346118792546956940632, −4.41955874685816877821382920981, −3.67659661269898197194758563403, −3.03538156105061200759572949400, −1.70965449751274279349505212042, −0.946094456404816744025198268194,
0.946094456404816744025198268194, 1.70965449751274279349505212042, 3.03538156105061200759572949400, 3.67659661269898197194758563403, 4.41955874685816877821382920981, 5.74985936346118792546956940632, 6.32891082747463643586427384833, 7.31715568476819843350858462741, 7.70107674785881678622186592031, 8.582903535103051671220586523393
