L(s) = 1 | − 3-s − 5-s + 9-s + 11-s − 4·13-s + 15-s − 2·17-s − 6·19-s + 4·23-s + 25-s − 27-s − 4·29-s − 33-s + 6·37-s + 4·39-s + 8·43-s − 45-s − 4·47-s + 2·51-s − 6·53-s − 55-s + 6·57-s + 4·59-s − 6·61-s + 4·65-s + 4·67-s − 4·69-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.447·5-s + 1/3·9-s + 0.301·11-s − 1.10·13-s + 0.258·15-s − 0.485·17-s − 1.37·19-s + 0.834·23-s + 1/5·25-s − 0.192·27-s − 0.742·29-s − 0.174·33-s + 0.986·37-s + 0.640·39-s + 1.21·43-s − 0.149·45-s − 0.583·47-s + 0.280·51-s − 0.824·53-s − 0.134·55-s + 0.794·57-s + 0.520·59-s − 0.768·61-s + 0.496·65-s + 0.488·67-s − 0.481·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 129360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 129360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8860675339\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8860675339\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
good | 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 - 6 T + p T^{2} \) |
| 83 | \( 1 - 14 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−13.24204073440510, −12.91346730678832, −12.59337141694665, −12.00950113596234, −11.55872617641950, −11.10171770283934, −10.67076372995698, −10.26675653392073, −9.450877547975000, −9.258690147257595, −8.660587105575865, −7.921900131151613, −7.617450537692588, −7.007104603391624, −6.505653446698833, −6.126145559923680, −5.399117063295818, −4.799514948112729, −4.477996570885219, −3.890552297259538, −3.235962457487695, −2.421642342435235, −2.022132835521797, −1.067038929681714, −0.3275859091098580,
0.3275859091098580, 1.067038929681714, 2.022132835521797, 2.421642342435235, 3.235962457487695, 3.890552297259538, 4.477996570885219, 4.799514948112729, 5.399117063295818, 6.126145559923680, 6.505653446698833, 7.007104603391624, 7.617450537692588, 7.921900131151613, 8.660587105575865, 9.258690147257595, 9.450877547975000, 10.26675653392073, 10.67076372995698, 11.10171770283934, 11.55872617641950, 12.00950113596234, 12.59337141694665, 12.91346730678832, 13.24204073440510