L(s) = 1 | + 1.15·3-s − 5-s − 0.465·7-s − 1.65·9-s − 2.23·11-s + 4.62·13-s − 1.15·15-s − 5.79·17-s + 8.52·19-s − 0.539·21-s + 23-s + 25-s − 5.39·27-s − 0.849·29-s + 0.420·31-s − 2.58·33-s + 0.465·35-s − 8.02·37-s + 5.35·39-s + 9.83·41-s − 10.7·43-s + 1.65·45-s + 9.83·47-s − 6.78·49-s − 6.70·51-s − 0.117·53-s + 2.23·55-s + ⋯ |
L(s) = 1 | + 0.668·3-s − 0.447·5-s − 0.176·7-s − 0.553·9-s − 0.672·11-s + 1.28·13-s − 0.298·15-s − 1.40·17-s + 1.95·19-s − 0.117·21-s + 0.208·23-s + 0.200·25-s − 1.03·27-s − 0.157·29-s + 0.0755·31-s − 0.449·33-s + 0.0787·35-s − 1.31·37-s + 0.858·39-s + 1.53·41-s − 1.64·43-s + 0.247·45-s + 1.43·47-s − 0.969·49-s − 0.938·51-s − 0.0161·53-s + 0.300·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 23 | \( 1 - T \) |
good | 3 | \( 1 - 1.15T + 3T^{2} \) |
| 7 | \( 1 + 0.465T + 7T^{2} \) |
| 11 | \( 1 + 2.23T + 11T^{2} \) |
| 13 | \( 1 - 4.62T + 13T^{2} \) |
| 17 | \( 1 + 5.79T + 17T^{2} \) |
| 19 | \( 1 - 8.52T + 19T^{2} \) |
| 29 | \( 1 + 0.849T + 29T^{2} \) |
| 31 | \( 1 - 0.420T + 31T^{2} \) |
| 37 | \( 1 + 8.02T + 37T^{2} \) |
| 41 | \( 1 - 9.83T + 41T^{2} \) |
| 43 | \( 1 + 10.7T + 43T^{2} \) |
| 47 | \( 1 - 9.83T + 47T^{2} \) |
| 53 | \( 1 + 0.117T + 53T^{2} \) |
| 59 | \( 1 + 5.74T + 59T^{2} \) |
| 61 | \( 1 - 11.1T + 61T^{2} \) |
| 67 | \( 1 + 13.0T + 67T^{2} \) |
| 71 | \( 1 + 5.33T + 71T^{2} \) |
| 73 | \( 1 + 10.3T + 73T^{2} \) |
| 79 | \( 1 - 2.81T + 79T^{2} \) |
| 83 | \( 1 - 4.53T + 83T^{2} \) |
| 89 | \( 1 + 2.72T + 89T^{2} \) |
| 97 | \( 1 + 9.33T + 97T^{2} \) |
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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
−7.60966443491195266373556746472, −7.04571457284796700618185211076, −6.12463183680501213801580850769, −5.48386792343980834610901637908, −4.66957250302024928543281419305, −3.70514679746725693954527107776, −3.19208157220044063744368992958, −2.45275419774164527456428259527, −1.31838109981693287303710392742, 0,
1.31838109981693287303710392742, 2.45275419774164527456428259527, 3.19208157220044063744368992958, 3.70514679746725693954527107776, 4.66957250302024928543281419305, 5.48386792343980834610901637908, 6.12463183680501213801580850769, 7.04571457284796700618185211076, 7.60966443491195266373556746472