| L(s) = 1 | − 2.31e8·3-s − 1.08e13·5-s − 4.54e15·7-s − 3.96e17·9-s + 1.61e19·11-s − 5.66e20·13-s + 2.50e21·15-s − 2.03e21·17-s + 2.44e22·19-s + 1.05e24·21-s − 1.30e25·23-s + 4.39e25·25-s + 1.96e26·27-s − 1.16e27·29-s + 6.59e27·31-s − 3.74e27·33-s + 4.91e28·35-s + 1.44e29·37-s + 1.31e29·39-s + 1.04e30·41-s + 2.02e30·43-s + 4.28e30·45-s − 7.03e30·47-s + 2.13e30·49-s + 4.71e29·51-s − 1.24e32·53-s − 1.74e32·55-s + ⋯ |
| L(s) = 1 | − 0.345·3-s − 1.26·5-s − 1.05·7-s − 0.880·9-s + 0.877·11-s − 1.39·13-s + 0.437·15-s − 0.0350·17-s + 0.0538·19-s + 0.364·21-s − 0.838·23-s + 0.603·25-s + 0.649·27-s − 1.03·29-s + 1.69·31-s − 0.303·33-s + 1.33·35-s + 1.41·37-s + 0.482·39-s + 1.52·41-s + 1.22·43-s + 1.11·45-s − 0.819·47-s + 0.115·49-s + 0.0121·51-s − 1.56·53-s − 1.11·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(38-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s+37/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(19)\) |
\(\approx\) |
\(0.5338989841\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5338989841\) |
| \(L(\frac{39}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| good | 3 | \( 1 + 2.31e8T + 4.50e17T^{2} \) |
| 5 | \( 1 + 1.08e13T + 7.27e25T^{2} \) |
| 7 | \( 1 + 4.54e15T + 1.85e31T^{2} \) |
| 11 | \( 1 - 1.61e19T + 3.40e38T^{2} \) |
| 13 | \( 1 + 5.66e20T + 1.64e41T^{2} \) |
| 17 | \( 1 + 2.03e21T + 3.36e45T^{2} \) |
| 19 | \( 1 - 2.44e22T + 2.06e47T^{2} \) |
| 23 | \( 1 + 1.30e25T + 2.42e50T^{2} \) |
| 29 | \( 1 + 1.16e27T + 1.28e54T^{2} \) |
| 31 | \( 1 - 6.59e27T + 1.51e55T^{2} \) |
| 37 | \( 1 - 1.44e29T + 1.05e58T^{2} \) |
| 41 | \( 1 - 1.04e30T + 4.70e59T^{2} \) |
| 43 | \( 1 - 2.02e30T + 2.74e60T^{2} \) |
| 47 | \( 1 + 7.03e30T + 7.37e61T^{2} \) |
| 53 | \( 1 + 1.24e32T + 6.28e63T^{2} \) |
| 59 | \( 1 + 3.71e32T + 3.32e65T^{2} \) |
| 61 | \( 1 + 2.11e32T + 1.14e66T^{2} \) |
| 67 | \( 1 + 3.63e33T + 3.67e67T^{2} \) |
| 71 | \( 1 - 3.36e34T + 3.13e68T^{2} \) |
| 73 | \( 1 + 3.78e34T + 8.76e68T^{2} \) |
| 79 | \( 1 + 1.82e35T + 1.63e70T^{2} \) |
| 83 | \( 1 + 1.67e35T + 1.01e71T^{2} \) |
| 89 | \( 1 - 9.60e35T + 1.34e72T^{2} \) |
| 97 | \( 1 - 6.57e36T + 3.24e73T^{2} \) |
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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
−16.00411018906582960616844651935, −14.52278012656763544534107129781, −12.42064234849410695909694218733, −11.44209189065042373294717436936, −9.517583005217942127665683938484, −7.74771658978220897600512942809, −6.22198621903759469765628983961, −4.29506598830855734989018877754, −2.88183590501335214989437136216, −0.42975724129345827606476198698,
0.42975724129345827606476198698, 2.88183590501335214989437136216, 4.29506598830855734989018877754, 6.22198621903759469765628983961, 7.74771658978220897600512942809, 9.517583005217942127665683938484, 11.44209189065042373294717436936, 12.42064234849410695909694218733, 14.52278012656763544534107129781, 16.00411018906582960616844651935
