| L(s) = 1 | − 27·3-s − 230.·5-s + 343·7-s + 729·9-s − 5.02e3·11-s − 1.37e4·13-s + 6.22e3·15-s + 3.24e4·17-s + 9.65e3·19-s − 9.26e3·21-s + 3.20e4·23-s − 2.50e4·25-s − 1.96e4·27-s + 1.03e5·29-s + 2.41e5·31-s + 1.35e5·33-s − 7.90e4·35-s − 1.27e5·37-s + 3.71e5·39-s + 6.07e5·41-s + 4.43e5·43-s − 1.67e5·45-s + 6.91e5·47-s + 1.17e5·49-s − 8.76e5·51-s + 6.72e5·53-s + 1.15e6·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.824·5-s + 0.377·7-s + 0.333·9-s − 1.13·11-s − 1.73·13-s + 0.475·15-s + 1.60·17-s + 0.322·19-s − 0.218·21-s + 0.548·23-s − 0.320·25-s − 0.192·27-s + 0.789·29-s + 1.45·31-s + 0.657·33-s − 0.311·35-s − 0.412·37-s + 1.00·39-s + 1.37·41-s + 0.851·43-s − 0.274·45-s + 0.971·47-s + 0.142·49-s − 0.925·51-s + 0.620·53-s + 0.938·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.075219392\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.075219392\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 27T \) |
| 7 | \( 1 - 343T \) |
| good | 5 | \( 1 + 230.T + 7.81e4T^{2} \) |
| 11 | \( 1 + 5.02e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 1.37e4T + 6.27e7T^{2} \) |
| 17 | \( 1 - 3.24e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 9.65e3T + 8.93e8T^{2} \) |
| 23 | \( 1 - 3.20e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 1.03e5T + 1.72e10T^{2} \) |
| 31 | \( 1 - 2.41e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 1.27e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 6.07e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 4.43e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 6.91e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 6.72e5T + 1.17e12T^{2} \) |
| 59 | \( 1 + 2.58e6T + 2.48e12T^{2} \) |
| 61 | \( 1 - 1.53e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 4.20e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 1.51e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 5.47e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 6.75e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 8.36e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 5.17e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 1.06e7T + 8.07e13T^{2} \) |
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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
−12.41572301081568519032734679827, −11.98280634167297158135393728015, −10.69411363917267662296796304112, −9.759871594598183547888901563070, −7.957714055616341217747331968084, −7.34609027537248213398591864923, −5.53290838544673194859807209315, −4.54542822684898988517015776932, −2.76767638564978787478410188387, −0.67960444434266820959569190865,
0.67960444434266820959569190865, 2.76767638564978787478410188387, 4.54542822684898988517015776932, 5.53290838544673194859807209315, 7.34609027537248213398591864923, 7.957714055616341217747331968084, 9.759871594598183547888901563070, 10.69411363917267662296796304112, 11.98280634167297158135393728015, 12.41572301081568519032734679827
