L(s) = 1 | + 2·3-s + 7-s + 9-s − 4·11-s + 6·13-s + 2·21-s − 5·25-s − 4·27-s + 2·29-s + 2·31-s − 8·33-s − 2·37-s + 12·39-s − 6·41-s − 4·43-s − 2·47-s + 49-s − 14·53-s + 14·59-s − 12·61-s + 63-s + 4·67-s − 2·73-s − 10·75-s − 4·77-s + 8·79-s − 11·81-s + ⋯ |
L(s) = 1 | + 1.15·3-s + 0.377·7-s + 1/3·9-s − 1.20·11-s + 1.66·13-s + 0.436·21-s − 25-s − 0.769·27-s + 0.371·29-s + 0.359·31-s − 1.39·33-s − 0.328·37-s + 1.92·39-s − 0.937·41-s − 0.609·43-s − 0.291·47-s + 1/7·49-s − 1.92·53-s + 1.82·59-s − 1.53·61-s + 0.125·63-s + 0.488·67-s − 0.234·73-s − 1.15·75-s − 0.455·77-s + 0.900·79-s − 1.22·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 29624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29624 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 \) |
good | 3 | \( 1 - 2 T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 + 14 T + p T^{2} \) |
| 59 | \( 1 - 14 T + p T^{2} \) |
| 61 | \( 1 + 12 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 16 T + p T^{2} \) |
| 97 | \( 1 + 4 T + p T^{2} \) |
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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
−15.32680898973725, −14.97510259328496, −14.23947011587956, −13.77958073152971, −13.44318062373037, −13.04068304690376, −12.32770905990990, −11.50858020735945, −11.23731573851200, −10.43639889884307, −10.10332834937405, −9.310673914379908, −8.788482640326772, −8.250468332268896, −7.969624226724445, −7.441849710869402, −6.477562403078117, −6.036122323518407, −5.229888064978847, −4.698813527683850, −3.661214046796713, −3.481664669433315, −2.629427105179158, −1.995499944335339, −1.264194563770882, 0,
1.264194563770882, 1.995499944335339, 2.629427105179158, 3.481664669433315, 3.661214046796713, 4.698813527683850, 5.229888064978847, 6.036122323518407, 6.477562403078117, 7.441849710869402, 7.969624226724445, 8.250468332268896, 8.788482640326772, 9.310673914379908, 10.10332834937405, 10.43639889884307, 11.23731573851200, 11.50858020735945, 12.32770905990990, 13.04068304690376, 13.44318062373037, 13.77958073152971, 14.23947011587956, 14.97510259328496, 15.32680898973725