| L(s) = 1 | − 3-s − 2.92·5-s + 7-s + 9-s − 4.85·11-s − 2.55·13-s + 2.92·15-s + 1.63·17-s + 19-s − 21-s + 23-s + 3.55·25-s − 27-s − 9.48·29-s + 0.367·31-s + 4.85·33-s − 2.92·35-s + 2.21·37-s + 2.55·39-s + 11.9·41-s + 2.70·43-s − 2.92·45-s − 2.48·47-s + 49-s − 1.63·51-s − 7.55·53-s + 14.1·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.30·5-s + 0.377·7-s + 0.333·9-s − 1.46·11-s − 0.709·13-s + 0.755·15-s + 0.395·17-s + 0.229·19-s − 0.218·21-s + 0.208·23-s + 0.711·25-s − 0.192·27-s − 1.76·29-s + 0.0659·31-s + 0.844·33-s − 0.494·35-s + 0.364·37-s + 0.409·39-s + 1.86·41-s + 0.412·43-s − 0.436·45-s − 0.362·47-s + 0.142·49-s − 0.228·51-s − 1.03·53-s + 1.91·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1932 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1932 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7176438787\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7176438787\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| good | 5 | \( 1 + 2.92T + 5T^{2} \) |
| 11 | \( 1 + 4.85T + 11T^{2} \) |
| 13 | \( 1 + 2.55T + 13T^{2} \) |
| 17 | \( 1 - 1.63T + 17T^{2} \) |
| 19 | \( 1 - T + 19T^{2} \) |
| 29 | \( 1 + 9.48T + 29T^{2} \) |
| 31 | \( 1 - 0.367T + 31T^{2} \) |
| 37 | \( 1 - 2.21T + 37T^{2} \) |
| 41 | \( 1 - 11.9T + 41T^{2} \) |
| 43 | \( 1 - 2.70T + 43T^{2} \) |
| 47 | \( 1 + 2.48T + 47T^{2} \) |
| 53 | \( 1 + 7.55T + 53T^{2} \) |
| 59 | \( 1 - 7.55T + 59T^{2} \) |
| 61 | \( 1 - 9.92T + 61T^{2} \) |
| 67 | \( 1 + 2.92T + 67T^{2} \) |
| 71 | \( 1 + 7.82T + 71T^{2} \) |
| 73 | \( 1 - 7.48T + 73T^{2} \) |
| 79 | \( 1 - 12.0T + 79T^{2} \) |
| 83 | \( 1 - 9.48T + 83T^{2} \) |
| 89 | \( 1 - 1.44T + 89T^{2} \) |
| 97 | \( 1 + 4.74T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.235711478708338429458535306579, −8.042179937442035854778687444393, −7.70182970112176147682839463801, −7.11663943281566168223947502441, −5.82299576293647813849958181837, −5.12760248348539336263871135824, −4.37236893526299393076597343826, −3.42203905072613659478474853613, −2.27564886038201919212463350304, −0.56313659387788901936205849874,
0.56313659387788901936205849874, 2.27564886038201919212463350304, 3.42203905072613659478474853613, 4.37236893526299393076597343826, 5.12760248348539336263871135824, 5.82299576293647813849958181837, 7.11663943281566168223947502441, 7.70182970112176147682839463801, 8.042179937442035854778687444393, 9.235711478708338429458535306579
