L(s) = 1 | − 3-s − 5-s + 7-s + 9-s − 2·11-s − 6·13-s + 15-s − 4·17-s + 4·19-s − 21-s − 8·23-s − 4·25-s − 27-s − 2·29-s + 5·31-s + 2·33-s − 35-s + 3·37-s + 6·39-s − 6·41-s + 8·43-s − 45-s − 3·47-s − 6·49-s + 4·51-s − 7·53-s + 2·55-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.447·5-s + 0.377·7-s + 1/3·9-s − 0.603·11-s − 1.66·13-s + 0.258·15-s − 0.970·17-s + 0.917·19-s − 0.218·21-s − 1.66·23-s − 4/5·25-s − 0.192·27-s − 0.371·29-s + 0.898·31-s + 0.348·33-s − 0.169·35-s + 0.493·37-s + 0.960·39-s − 0.937·41-s + 1.21·43-s − 0.149·45-s − 0.437·47-s − 6/7·49-s + 0.560·51-s − 0.961·53-s + 0.269·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6315891867\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6315891867\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 167 | \( 1 + T \) |
good | 5 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 + 7 T + p T^{2} \) |
| 59 | \( 1 + 11 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 - 7 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 15 T + p T^{2} \) |
| 89 | \( 1 - 9 T + p T^{2} \) |
| 97 | \( 1 + 13 T + p T^{2} \) |
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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
−7.88336866383763772002101377288, −7.23101023617400228399309691381, −6.44230508244087771179361688871, −5.71481975225693688854554474233, −4.89846212681801535505883503585, −4.53995408291162478677996862365, −3.61174350260285628566901840168, −2.54308773765556490445671944309, −1.84441375551827396300217727287, −0.38626004032416677540888728569,
0.38626004032416677540888728569, 1.84441375551827396300217727287, 2.54308773765556490445671944309, 3.61174350260285628566901840168, 4.53995408291162478677996862365, 4.89846212681801535505883503585, 5.71481975225693688854554474233, 6.44230508244087771179361688871, 7.23101023617400228399309691381, 7.88336866383763772002101377288