L(s) = 1 | + 0.955·2-s + 1.79·3-s − 0.0871·4-s − 1.33·5-s + 1.71·6-s + 1.90·7-s − 1.03·8-s + 2.21·9-s − 1.27·10-s − 1.54·11-s − 0.156·12-s + 1.82·14-s − 2.38·15-s − 0.905·16-s + 2.11·18-s + 0.115·20-s + 3.41·21-s − 1.47·22-s − 1.86·24-s + 0.770·25-s + 2.17·27-s − 0.166·28-s + 0.676·29-s − 2.27·30-s + 0.173·32-s − 2.76·33-s − 2.53·35-s + ⋯ |
L(s) = 1 | + 0.955·2-s + 1.79·3-s − 0.0871·4-s − 1.33·5-s + 1.71·6-s + 1.90·7-s − 1.03·8-s + 2.21·9-s − 1.27·10-s − 1.54·11-s − 0.156·12-s + 1.82·14-s − 2.38·15-s − 0.905·16-s + 2.11·18-s + 0.115·20-s + 3.41·21-s − 1.47·22-s − 1.86·24-s + 0.770·25-s + 2.17·27-s − 0.166·28-s + 0.676·29-s − 2.27·30-s + 0.173·32-s − 2.76·33-s − 2.53·35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.291408021\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.291408021\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 1151 | \( 1 - T \) |
good | 2 | \( 1 - 0.955T + T^{2} \) |
| 3 | \( 1 - 1.79T + T^{2} \) |
| 5 | \( 1 + 1.33T + T^{2} \) |
| 7 | \( 1 - 1.90T + T^{2} \) |
| 11 | \( 1 + 1.54T + T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 - 0.676T + T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 + 1.71T + T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 + 1.99T + T^{2} \) |
| 47 | \( 1 + 1.94T + T^{2} \) |
| 53 | \( 1 - 0.0766T + T^{2} \) |
| 59 | \( 1 + 0.229T + T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 - 1.44T + T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 - 1.63T + T^{2} \) |
| 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
−9.940337485704121189587457490920, −8.679540308875644723427712373864, −8.225264337175558609772766189003, −7.934828455138790380756632730084, −7.00822060879165248225929562154, −5.07219976802220001600443484546, −4.76683768055546751946363478660, −3.78608654141864889910124530260, −3.08545181628854815361095443223, −1.97597329520505202385499367031,
1.97597329520505202385499367031, 3.08545181628854815361095443223, 3.78608654141864889910124530260, 4.76683768055546751946363478660, 5.07219976802220001600443484546, 7.00822060879165248225929562154, 7.934828455138790380756632730084, 8.225264337175558609772766189003, 8.679540308875644723427712373864, 9.940337485704121189587457490920