| L(s) = 1 | − 2-s + 3-s + 4-s − 5-s − 6-s + 2·7-s − 8-s + 9-s + 10-s + 12-s + 13-s − 2·14-s − 15-s + 16-s − 6·17-s − 18-s + 2·19-s − 20-s + 2·21-s + 3·23-s − 24-s + 25-s − 26-s + 27-s + 2·28-s + 30-s + 5·31-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s + 0.755·7-s − 0.353·8-s + 1/3·9-s + 0.316·10-s + 0.288·12-s + 0.277·13-s − 0.534·14-s − 0.258·15-s + 1/4·16-s − 1.45·17-s − 0.235·18-s + 0.458·19-s − 0.223·20-s + 0.436·21-s + 0.625·23-s − 0.204·24-s + 1/5·25-s − 0.196·26-s + 0.192·27-s + 0.377·28-s + 0.182·30-s + 0.898·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 327990 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 327990 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.520521907\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.520521907\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 29 | \( 1 \) |
| good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 - 2 T + p T^{2} \) |
| 83 | \( 1 - 3 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−12.64764997700626, −12.01433885334237, −11.40267823585213, −11.32798149853390, −10.89858649781760, −10.23138756911491, −9.833733871004233, −9.321736901295883, −8.899313052653279, −8.364355421527973, −8.088561716740419, −7.820247962897629, −6.985756554173421, −6.717665552051117, −6.387422694388742, −5.441043088736590, −5.002244654921933, −4.520940686196958, −3.983420430316928, −3.298254327723849, −2.945037042623153, −2.070346121628825, −1.900707567127676, −1.030004567659519, −0.5028493006475713,
0.5028493006475713, 1.030004567659519, 1.900707567127676, 2.070346121628825, 2.945037042623153, 3.298254327723849, 3.983420430316928, 4.520940686196958, 5.002244654921933, 5.441043088736590, 6.387422694388742, 6.717665552051117, 6.985756554173421, 7.820247962897629, 8.088561716740419, 8.364355421527973, 8.899313052653279, 9.321736901295883, 9.833733871004233, 10.23138756911491, 10.89858649781760, 11.32798149853390, 11.40267823585213, 12.01433885334237, 12.64764997700626
