| L(s) = 1 | + 2.32·2-s − 4.43·3-s − 2.57·4-s + 20.5·5-s − 10.3·6-s − 30.5·7-s − 24.6·8-s − 7.29·9-s + 47.8·10-s − 66.2·11-s + 11.4·12-s + 27.5·13-s − 71.2·14-s − 91.1·15-s − 36.7·16-s + 108.·17-s − 16.9·18-s + 70.9·19-s − 52.8·20-s + 135.·21-s − 154.·22-s + 23·23-s + 109.·24-s + 296.·25-s + 64.2·26-s + 152.·27-s + 78.7·28-s + ⋯ |
| L(s) = 1 | + 0.823·2-s − 0.854·3-s − 0.321·4-s + 1.83·5-s − 0.703·6-s − 1.65·7-s − 1.08·8-s − 0.270·9-s + 1.51·10-s − 1.81·11-s + 0.275·12-s + 0.588·13-s − 1.35·14-s − 1.56·15-s − 0.574·16-s + 1.55·17-s − 0.222·18-s + 0.856·19-s − 0.591·20-s + 1.41·21-s − 1.49·22-s + 0.208·23-s + 0.929·24-s + 2.37·25-s + 0.484·26-s + 1.08·27-s + 0.531·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 667 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 667 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.668338262\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.668338262\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 23 | \( 1 - 23T \) |
| 29 | \( 1 - 29T \) |
| good | 2 | \( 1 - 2.32T + 8T^{2} \) |
| 3 | \( 1 + 4.43T + 27T^{2} \) |
| 5 | \( 1 - 20.5T + 125T^{2} \) |
| 7 | \( 1 + 30.5T + 343T^{2} \) |
| 11 | \( 1 + 66.2T + 1.33e3T^{2} \) |
| 13 | \( 1 - 27.5T + 2.19e3T^{2} \) |
| 17 | \( 1 - 108.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 70.9T + 6.85e3T^{2} \) |
| 31 | \( 1 + 95.1T + 2.97e4T^{2} \) |
| 37 | \( 1 + 231.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 323.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 223.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 426.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 104.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 640.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 216.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 539.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 972.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 377.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 526.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 479.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.60e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 82.8T + 9.12e5T^{2} \) |
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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
−10.05292088664105481564599801014, −9.605893914246799374471931612433, −8.607385447635840547511326971724, −7.01184690572666878068733162828, −5.88411366294835055864166687483, −5.71026205072266914261528703670, −5.13337301526625440358978113575, −3.31294567266436740808038354943, −2.68125526266431290570841875347, −0.67318094350276175530027796620,
0.67318094350276175530027796620, 2.68125526266431290570841875347, 3.31294567266436740808038354943, 5.13337301526625440358978113575, 5.71026205072266914261528703670, 5.88411366294835055864166687483, 7.01184690572666878068733162828, 8.607385447635840547511326971724, 9.605893914246799374471931612433, 10.05292088664105481564599801014
