| L(s) = 1 | − 1.87·2-s + 1.28·3-s + 2.53·4-s + 0.684·5-s − 2.41·6-s + 7-s − 2.87·8-s + 0.652·9-s − 1.28·10-s − 11-s + 3.25·12-s + 1.73·13-s − 1.87·14-s + 0.879·15-s + 2.87·16-s + 1.96·17-s − 1.22·18-s + 1.73·20-s + 1.28·21-s + 1.87·22-s + 23-s − 3.70·24-s − 0.532·25-s − 3.25·26-s − 0.446·27-s + 2.53·28-s − 1.53·29-s + ⋯ |
| L(s) = 1 | − 1.87·2-s + 1.28·3-s + 2.53·4-s + 0.684·5-s − 2.41·6-s + 7-s − 2.87·8-s + 0.652·9-s − 1.28·10-s − 11-s + 3.25·12-s + 1.73·13-s − 1.87·14-s + 0.879·15-s + 2.87·16-s + 1.96·17-s − 1.22·18-s + 1.73·20-s + 1.28·21-s + 1.87·22-s + 23-s − 3.70·24-s − 0.532·25-s − 3.25·26-s − 0.446·27-s + 2.53·28-s − 1.53·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3311 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3311 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.138350640\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.138350640\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 43 | \( 1 + T \) |
| good | 2 | \( 1 + 1.87T + T^{2} \) |
| 3 | \( 1 - 1.28T + T^{2} \) |
| 5 | \( 1 - 0.684T + T^{2} \) |
| 13 | \( 1 - 1.73T + T^{2} \) |
| 17 | \( 1 - 1.96T + T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 - T + T^{2} \) |
| 29 | \( 1 + 1.53T + T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 + 1.96T + T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 - 0.347T + T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 + 1.87T + T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 + 1.28T + T^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−8.682067653566081650652785743755, −8.260377672667134108179664494165, −7.74548903643754415809456014338, −7.14939734691053601066042369649, −5.93243778724282680670798791433, −5.37347707111470368586468044400, −3.56530568524333269975747245730, −2.92538529423092698572150116760, −1.82551852563065318804379907044, −1.39371149546944267292698012476,
1.39371149546944267292698012476, 1.82551852563065318804379907044, 2.92538529423092698572150116760, 3.56530568524333269975747245730, 5.37347707111470368586468044400, 5.93243778724282680670798791433, 7.14939734691053601066042369649, 7.74548903643754415809456014338, 8.260377672667134108179664494165, 8.682067653566081650652785743755
