| L(s) = 1 | + 15.8·2-s + 41.5·3-s + 124.·4-s + 303.·5-s + 660.·6-s + 157.·7-s − 57.6·8-s − 458.·9-s + 4.82e3·10-s + 1.05e3·11-s + 5.17e3·12-s + 1.86e3·13-s + 2.49e3·14-s + 1.26e4·15-s − 1.68e4·16-s + 6.11e3·17-s − 7.29e3·18-s − 6.36e3·19-s + 3.77e4·20-s + 6.53e3·21-s + 1.68e4·22-s − 2.64e3·23-s − 2.39e3·24-s + 1.40e4·25-s + 2.96e4·26-s − 1.09e5·27-s + 1.95e4·28-s + ⋯ |
| L(s) = 1 | + 1.40·2-s + 0.888·3-s + 0.971·4-s + 1.08·5-s + 1.24·6-s + 0.173·7-s − 0.0398·8-s − 0.209·9-s + 1.52·10-s + 0.239·11-s + 0.863·12-s + 0.235·13-s + 0.243·14-s + 0.965·15-s − 1.02·16-s + 0.301·17-s − 0.294·18-s − 0.212·19-s + 1.05·20-s + 0.154·21-s + 0.336·22-s − 0.0453·23-s − 0.0354·24-s + 0.179·25-s + 0.330·26-s − 1.07·27-s + 0.168·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(5.062039513\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.062039513\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 37 | \( 1 + 5.06e4T \) |
| good | 2 | \( 1 - 15.8T + 128T^{2} \) |
| 3 | \( 1 - 41.5T + 2.18e3T^{2} \) |
| 5 | \( 1 - 303.T + 7.81e4T^{2} \) |
| 7 | \( 1 - 157.T + 8.23e5T^{2} \) |
| 11 | \( 1 - 1.05e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 1.86e3T + 6.27e7T^{2} \) |
| 17 | \( 1 - 6.11e3T + 4.10e8T^{2} \) |
| 19 | \( 1 + 6.36e3T + 8.93e8T^{2} \) |
| 23 | \( 1 + 2.64e3T + 3.40e9T^{2} \) |
| 29 | \( 1 - 4.07e4T + 1.72e10T^{2} \) |
| 31 | \( 1 - 7.14e4T + 2.75e10T^{2} \) |
| 41 | \( 1 + 6.68e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 1.77e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 6.49e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 9.34e5T + 1.17e12T^{2} \) |
| 59 | \( 1 - 6.78e5T + 2.48e12T^{2} \) |
| 61 | \( 1 - 1.23e6T + 3.14e12T^{2} \) |
| 67 | \( 1 - 3.49e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 2.48e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 5.75e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 3.01e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 6.97e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 2.14e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 9.22e6T + 8.07e13T^{2} \) |
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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
−14.39581944239288910090368281302, −13.83373365446751579697501975732, −12.95814955618134506268449970480, −11.56129369399906885292666253881, −9.789089697657103552114280265792, −8.489253236460487355402932252022, −6.44970947359615171929622906129, −5.21280295136437613177734057940, −3.49739914711234879654348896596, −2.14923808116560336202324711305,
2.14923808116560336202324711305, 3.49739914711234879654348896596, 5.21280295136437613177734057940, 6.44970947359615171929622906129, 8.489253236460487355402932252022, 9.789089697657103552114280265792, 11.56129369399906885292666253881, 12.95814955618134506268449970480, 13.83373365446751579697501975732, 14.39581944239288910090368281302
