L(s) = 1 | − 3-s − 4.07·5-s − 7-s + 9-s + 6.53·11-s + 4.76·13-s + 4.07·15-s + 2.68·17-s + 4.83·19-s + 21-s + 23-s + 11.6·25-s − 27-s + 7.88·29-s + 0.685·31-s − 6.53·33-s + 4.07·35-s − 7.88·37-s − 4.76·39-s − 7.88·41-s + 4.61·43-s − 4.07·45-s + 10.5·47-s + 49-s − 2.68·51-s + 8.19·53-s − 26.6·55-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 1.82·5-s − 0.377·7-s + 0.333·9-s + 1.97·11-s + 1.32·13-s + 1.05·15-s + 0.651·17-s + 1.10·19-s + 0.218·21-s + 0.208·23-s + 2.32·25-s − 0.192·27-s + 1.46·29-s + 0.123·31-s − 1.13·33-s + 0.688·35-s − 1.29·37-s − 0.762·39-s − 1.23·41-s + 0.703·43-s − 0.607·45-s + 1.54·47-s + 0.142·49-s − 0.376·51-s + 1.12·53-s − 3.59·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.532573431\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.532573431\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 - T \) |
good | 5 | \( 1 + 4.07T + 5T^{2} \) |
| 11 | \( 1 - 6.53T + 11T^{2} \) |
| 13 | \( 1 - 4.76T + 13T^{2} \) |
| 17 | \( 1 - 2.68T + 17T^{2} \) |
| 19 | \( 1 - 4.83T + 19T^{2} \) |
| 29 | \( 1 - 7.88T + 29T^{2} \) |
| 31 | \( 1 - 0.685T + 31T^{2} \) |
| 37 | \( 1 + 7.88T + 37T^{2} \) |
| 41 | \( 1 + 7.88T + 41T^{2} \) |
| 43 | \( 1 - 4.61T + 43T^{2} \) |
| 47 | \( 1 - 10.5T + 47T^{2} \) |
| 53 | \( 1 - 8.19T + 53T^{2} \) |
| 59 | \( 1 + 5.42T + 59T^{2} \) |
| 61 | \( 1 - 5.80T + 61T^{2} \) |
| 67 | \( 1 + 2.72T + 67T^{2} \) |
| 71 | \( 1 + 0.342T + 71T^{2} \) |
| 73 | \( 1 - 12.6T + 73T^{2} \) |
| 79 | \( 1 + 6.53T + 79T^{2} \) |
| 83 | \( 1 + 11.4T + 83T^{2} \) |
| 89 | \( 1 - 7.51T + 89T^{2} \) |
| 97 | \( 1 - 18.1T + 97T^{2} \) |
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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
−7.77766068491896113539814896415, −7.02781705298556534420281931347, −6.67602130113335688054277916779, −5.87036375937495665687531168807, −4.96002611743273168719115273239, −4.04615392213019867465680190949, −3.73641375688454048038509341030, −3.09469632353544560134861995370, −1.27303659523327064095789331848, −0.77122110585245585765335016916,
0.77122110585245585765335016916, 1.27303659523327064095789331848, 3.09469632353544560134861995370, 3.73641375688454048038509341030, 4.04615392213019867465680190949, 4.96002611743273168719115273239, 5.87036375937495665687531168807, 6.67602130113335688054277916779, 7.02781705298556534420281931347, 7.77766068491896113539814896415