L(s) = 1 | − 2-s + 4-s + 2.87·5-s − 0.652·7-s − 8-s − 2.87·10-s − 3.41·11-s − 1.81·13-s + 0.652·14-s + 16-s − 3.63·17-s − 2.53·19-s + 2.87·20-s + 3.41·22-s + 6.10·23-s + 3.29·25-s + 1.81·26-s − 0.652·28-s + 6.78·29-s − 5.92·31-s − 32-s + 3.63·34-s − 1.87·35-s − 7.08·37-s + 2.53·38-s − 2.87·40-s + 4.92·41-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.5·4-s + 1.28·5-s − 0.246·7-s − 0.353·8-s − 0.910·10-s − 1.02·11-s − 0.503·13-s + 0.174·14-s + 0.250·16-s − 0.882·17-s − 0.580·19-s + 0.643·20-s + 0.727·22-s + 1.27·23-s + 0.658·25-s + 0.355·26-s − 0.123·28-s + 1.25·29-s − 1.06·31-s − 0.176·32-s + 0.623·34-s − 0.317·35-s − 1.16·37-s + 0.410·38-s − 0.455·40-s + 0.768·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 223 | \( 1 - T \) |
good | 5 | \( 1 - 2.87T + 5T^{2} \) |
| 7 | \( 1 + 0.652T + 7T^{2} \) |
| 11 | \( 1 + 3.41T + 11T^{2} \) |
| 13 | \( 1 + 1.81T + 13T^{2} \) |
| 17 | \( 1 + 3.63T + 17T^{2} \) |
| 19 | \( 1 + 2.53T + 19T^{2} \) |
| 23 | \( 1 - 6.10T + 23T^{2} \) |
| 29 | \( 1 - 6.78T + 29T^{2} \) |
| 31 | \( 1 + 5.92T + 31T^{2} \) |
| 37 | \( 1 + 7.08T + 37T^{2} \) |
| 41 | \( 1 - 4.92T + 41T^{2} \) |
| 43 | \( 1 - 4.59T + 43T^{2} \) |
| 47 | \( 1 - 12.3T + 47T^{2} \) |
| 53 | \( 1 + 9.80T + 53T^{2} \) |
| 59 | \( 1 - 0.573T + 59T^{2} \) |
| 61 | \( 1 + 9.10T + 61T^{2} \) |
| 67 | \( 1 - 1.85T + 67T^{2} \) |
| 71 | \( 1 + 3T + 71T^{2} \) |
| 73 | \( 1 + 1.42T + 73T^{2} \) |
| 79 | \( 1 + 2.71T + 79T^{2} \) |
| 83 | \( 1 + 13.8T + 83T^{2} \) |
| 89 | \( 1 + 12.6T + 89T^{2} \) |
| 97 | \( 1 + 13.5T + 97T^{2} \) |
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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
−8.221702254788877543038592582849, −7.27301114565281815207124495406, −6.73669820501435009875328852977, −5.90811604316174739294307818588, −5.28331182001181423615977909443, −4.40675880297888010775039975094, −2.92921110256464231050264437429, −2.42590670274807802506171257024, −1.46598680503735829090595980918, 0,
1.46598680503735829090595980918, 2.42590670274807802506171257024, 2.92921110256464231050264437429, 4.40675880297888010775039975094, 5.28331182001181423615977909443, 5.90811604316174739294307818588, 6.73669820501435009875328852977, 7.27301114565281815207124495406, 8.221702254788877543038592582849