| L(s) = 1 | − 15.0·2-s + 27·3-s + 97.4·4-s + 6.55·5-s − 405.·6-s + 1.46e3·7-s + 458.·8-s + 729·9-s − 98.3·10-s + 4.59e3·11-s + 2.63e3·12-s + 1.52e4·13-s − 2.20e4·14-s + 176.·15-s − 1.93e4·16-s + 3.89e4·17-s − 1.09e4·18-s − 499.·19-s + 638.·20-s + 3.95e4·21-s − 6.90e4·22-s + 5.82e4·23-s + 1.23e4·24-s − 7.80e4·25-s − 2.28e5·26-s + 1.96e4·27-s + 1.42e5·28-s + ⋯ |
| L(s) = 1 | − 1.32·2-s + 0.577·3-s + 0.761·4-s + 0.0234·5-s − 0.766·6-s + 1.61·7-s + 0.316·8-s + 0.333·9-s − 0.0311·10-s + 1.04·11-s + 0.439·12-s + 1.92·13-s − 2.14·14-s + 0.0135·15-s − 1.18·16-s + 1.92·17-s − 0.442·18-s − 0.0167·19-s + 0.0178·20-s + 0.932·21-s − 1.38·22-s + 0.998·23-s + 0.182·24-s − 0.999·25-s − 2.55·26-s + 0.192·27-s + 1.23·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(2.154221295\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.154221295\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 27T \) |
| 59 | \( 1 - 2.05e5T \) |
| good | 2 | \( 1 + 15.0T + 128T^{2} \) |
| 5 | \( 1 - 6.55T + 7.81e4T^{2} \) |
| 7 | \( 1 - 1.46e3T + 8.23e5T^{2} \) |
| 11 | \( 1 - 4.59e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 1.52e4T + 6.27e7T^{2} \) |
| 17 | \( 1 - 3.89e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 499.T + 8.93e8T^{2} \) |
| 23 | \( 1 - 5.82e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 2.11e5T + 1.72e10T^{2} \) |
| 31 | \( 1 - 6.18e4T + 2.75e10T^{2} \) |
| 37 | \( 1 + 6.29e4T + 9.49e10T^{2} \) |
| 41 | \( 1 + 6.19e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 3.56e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 9.53e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 1.97e6T + 1.17e12T^{2} \) |
| 61 | \( 1 + 2.98e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 1.04e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 4.10e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 2.60e5T + 1.10e13T^{2} \) |
| 79 | \( 1 + 1.33e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 8.30e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 3.49e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 1.00e7T + 8.07e13T^{2} \) |
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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
−11.15543730707037320303446967532, −10.23114839830096505879882573011, −9.167085213614107021945864648784, −8.259760164147606593780773430040, −7.943920646519697852069834320750, −6.50428894468619726509360889424, −4.85461455332262024116488799247, −3.49482336591809486354092833607, −1.42180224023623939772885336295, −1.27412100385318772932881929018,
1.27412100385318772932881929018, 1.42180224023623939772885336295, 3.49482336591809486354092833607, 4.85461455332262024116488799247, 6.50428894468619726509360889424, 7.943920646519697852069834320750, 8.259760164147606593780773430040, 9.167085213614107021945864648784, 10.23114839830096505879882573011, 11.15543730707037320303446967532
