| L(s) = 1 | − 1.71·2-s + 2.30·3-s + 0.930·4-s − 1.06·5-s − 3.95·6-s + 0.837·7-s + 1.83·8-s + 2.32·9-s + 1.82·10-s − 5.81·11-s + 2.14·12-s − 3.82·13-s − 1.43·14-s − 2.45·15-s − 4.99·16-s − 3.14·17-s − 3.98·18-s + 3.38·19-s − 0.991·20-s + 1.93·21-s + 9.95·22-s − 4.10·23-s + 4.22·24-s − 3.86·25-s + 6.55·26-s − 1.54·27-s + 0.779·28-s + ⋯ |
| L(s) = 1 | − 1.21·2-s + 1.33·3-s + 0.465·4-s − 0.476·5-s − 1.61·6-s + 0.316·7-s + 0.647·8-s + 0.776·9-s + 0.576·10-s − 1.75·11-s + 0.620·12-s − 1.06·13-s − 0.383·14-s − 0.634·15-s − 1.24·16-s − 0.762·17-s − 0.939·18-s + 0.776·19-s − 0.221·20-s + 0.421·21-s + 2.12·22-s − 0.856·23-s + 0.862·24-s − 0.773·25-s + 1.28·26-s − 0.298·27-s + 0.147·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9409 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9409 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6805167958\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6805167958\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 97 | \( 1 \) |
| good | 2 | \( 1 + 1.71T + 2T^{2} \) |
| 3 | \( 1 - 2.30T + 3T^{2} \) |
| 5 | \( 1 + 1.06T + 5T^{2} \) |
| 7 | \( 1 - 0.837T + 7T^{2} \) |
| 11 | \( 1 + 5.81T + 11T^{2} \) |
| 13 | \( 1 + 3.82T + 13T^{2} \) |
| 17 | \( 1 + 3.14T + 17T^{2} \) |
| 19 | \( 1 - 3.38T + 19T^{2} \) |
| 23 | \( 1 + 4.10T + 23T^{2} \) |
| 29 | \( 1 - 2.85T + 29T^{2} \) |
| 31 | \( 1 + 4.74T + 31T^{2} \) |
| 37 | \( 1 + 0.202T + 37T^{2} \) |
| 41 | \( 1 + 9.14T + 41T^{2} \) |
| 43 | \( 1 - 0.400T + 43T^{2} \) |
| 47 | \( 1 - 7.27T + 47T^{2} \) |
| 53 | \( 1 - 13.3T + 53T^{2} \) |
| 59 | \( 1 + 4.74T + 59T^{2} \) |
| 61 | \( 1 + 7.15T + 61T^{2} \) |
| 67 | \( 1 - 15.6T + 67T^{2} \) |
| 71 | \( 1 - 5.34T + 71T^{2} \) |
| 73 | \( 1 - 8.85T + 73T^{2} \) |
| 79 | \( 1 + 14.5T + 79T^{2} \) |
| 83 | \( 1 - 14.1T + 83T^{2} \) |
| 89 | \( 1 + 3.81T + 89T^{2} \) |
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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
−7.84592360379781598703898246669, −7.53049324826749430590303675483, −6.89009741845036339736255584845, −5.52833395363167186240240142112, −4.88082263624002793696939388538, −4.08939082498205183274590064174, −3.20553015028220946923295470457, −2.30440098843734934496566111670, −1.96238125366327340875349123065, −0.41744592544154408496808823271,
0.41744592544154408496808823271, 1.96238125366327340875349123065, 2.30440098843734934496566111670, 3.20553015028220946923295470457, 4.08939082498205183274590064174, 4.88082263624002793696939388538, 5.52833395363167186240240142112, 6.89009741845036339736255584845, 7.53049324826749430590303675483, 7.84592360379781598703898246669
