L(s) = 1 | − 2.57·3-s + 5-s − 7-s + 3.64·9-s − 11-s − 3.13·13-s − 2.57·15-s − 5.13·17-s − 8.04·19-s + 2.57·21-s − 5.21·23-s + 25-s − 1.67·27-s − 2.56·29-s − 4.48·31-s + 2.57·33-s − 35-s + 5.91·37-s + 8.09·39-s − 3.41·41-s − 0.831·43-s + 3.64·45-s − 11.9·47-s + 49-s + 13.2·51-s + 7.21·53-s − 55-s + ⋯ |
L(s) = 1 | − 1.48·3-s + 0.447·5-s − 0.377·7-s + 1.21·9-s − 0.301·11-s − 0.870·13-s − 0.665·15-s − 1.24·17-s − 1.84·19-s + 0.562·21-s − 1.08·23-s + 0.200·25-s − 0.322·27-s − 0.475·29-s − 0.806·31-s + 0.448·33-s − 0.169·35-s + 0.971·37-s + 1.29·39-s − 0.532·41-s − 0.126·43-s + 0.544·45-s − 1.73·47-s + 0.142·49-s + 1.85·51-s + 0.990·53-s − 0.134·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.2843394736\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2843394736\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 + T \) |
good | 3 | \( 1 + 2.57T + 3T^{2} \) |
| 13 | \( 1 + 3.13T + 13T^{2} \) |
| 17 | \( 1 + 5.13T + 17T^{2} \) |
| 19 | \( 1 + 8.04T + 19T^{2} \) |
| 23 | \( 1 + 5.21T + 23T^{2} \) |
| 29 | \( 1 + 2.56T + 29T^{2} \) |
| 31 | \( 1 + 4.48T + 31T^{2} \) |
| 37 | \( 1 - 5.91T + 37T^{2} \) |
| 41 | \( 1 + 3.41T + 41T^{2} \) |
| 43 | \( 1 + 0.831T + 43T^{2} \) |
| 47 | \( 1 + 11.9T + 47T^{2} \) |
| 53 | \( 1 - 7.21T + 53T^{2} \) |
| 59 | \( 1 - 6.11T + 59T^{2} \) |
| 61 | \( 1 - 4.48T + 61T^{2} \) |
| 67 | \( 1 + 4.50T + 67T^{2} \) |
| 71 | \( 1 - 14.4T + 71T^{2} \) |
| 73 | \( 1 + 1.17T + 73T^{2} \) |
| 79 | \( 1 + 4.69T + 79T^{2} \) |
| 83 | \( 1 + 9.75T + 83T^{2} \) |
| 89 | \( 1 - 12.7T + 89T^{2} \) |
| 97 | \( 1 + 9.35T + 97T^{2} \) |
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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
−8.031680200469811533564756987029, −6.94146785837682720467696586880, −6.60684563235315990892377779482, −5.94372772503949733003345132716, −5.30857469438157997113094391128, −4.57727798654404227030926199876, −3.94229470422583842443084276444, −2.52262583390764853787463095463, −1.84203880617151718546449199934, −0.28449687001947587335974994027,
0.28449687001947587335974994027, 1.84203880617151718546449199934, 2.52262583390764853787463095463, 3.94229470422583842443084276444, 4.57727798654404227030926199876, 5.30857469438157997113094391128, 5.94372772503949733003345132716, 6.60684563235315990892377779482, 6.94146785837682720467696586880, 8.031680200469811533564756987029