L(s) = 1 | − 0.347·2-s + 1.87·3-s − 0.879·4-s − 0.347·5-s − 0.652·6-s − 7-s + 0.652·8-s + 2.53·9-s + 0.120·10-s + 11-s − 1.65·12-s − 13-s + 0.347·14-s − 0.652·15-s + 0.652·16-s + 1.53·17-s − 0.879·18-s + 0.305·20-s − 1.87·21-s − 0.347·22-s − 23-s + 1.22·24-s − 0.879·25-s + 0.347·26-s + 2.87·27-s + 0.879·28-s + 1.87·29-s + ⋯ |
L(s) = 1 | − 0.347·2-s + 1.87·3-s − 0.879·4-s − 0.347·5-s − 0.652·6-s − 7-s + 0.652·8-s + 2.53·9-s + 0.120·10-s + 11-s − 1.65·12-s − 13-s + 0.347·14-s − 0.652·15-s + 0.652·16-s + 1.53·17-s − 0.879·18-s + 0.305·20-s − 1.87·21-s − 0.347·22-s − 23-s + 1.22·24-s − 0.879·25-s + 0.347·26-s + 2.87·27-s + 0.879·28-s + 1.87·29-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
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.560399688\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.560399688\) |
\(L(1)\) |
|
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 | \( 1 + 0.347T + T^{2} \) |
| 3 | \( 1 - 1.87T + T^{2} \) |
| 5 | \( 1 + 0.347T + T^{2} \) |
| 13 | \( 1 + T + T^{2} \) |
| 17 | \( 1 - 1.53T + T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 + T + T^{2} \) |
| 29 | \( 1 - 1.87T + T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 - 1.53T + T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 - 1.53T + T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 - 0.347T + T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 + 1.87T + T^{2} \) |
| 89 | \( 1 + 2T + T^{2} \) |
| 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
−8.740124512261000796613877837907, −8.257443163110442206096831423486, −7.56347084902920474100059323560, −7.00760170215587445860040121859, −5.85882397546256010053620974521, −4.55728651411710535468314231644, −3.92263838188277592935385883376, −3.34712125717025835726075998281, −2.45324313751879379526484172041, −1.15900893831949460014946500972,
1.15900893831949460014946500972, 2.45324313751879379526484172041, 3.34712125717025835726075998281, 3.92263838188277592935385883376, 4.55728651411710535468314231644, 5.85882397546256010053620974521, 7.00760170215587445860040121859, 7.56347084902920474100059323560, 8.257443163110442206096831423486, 8.740124512261000796613877837907