L(s) = 1 | − 2.22·3-s + 2.71·5-s − 1.08·7-s + 1.94·9-s − 3.45·11-s + 13-s − 6.03·15-s + 0.0669·17-s − 7.68·19-s + 2.40·21-s + 6.33·23-s + 2.37·25-s + 2.35·27-s + 29-s + 6.08·31-s + 7.67·33-s − 2.93·35-s + 5.46·37-s − 2.22·39-s + 5.02·41-s + 2.71·43-s + 5.26·45-s − 1.68·47-s − 5.83·49-s − 0.148·51-s − 8.77·53-s − 9.38·55-s + ⋯ |
L(s) = 1 | − 1.28·3-s + 1.21·5-s − 0.408·7-s + 0.646·9-s − 1.04·11-s + 0.277·13-s − 1.55·15-s + 0.0162·17-s − 1.76·19-s + 0.524·21-s + 1.32·23-s + 0.474·25-s + 0.453·27-s + 0.185·29-s + 1.09·31-s + 1.33·33-s − 0.496·35-s + 0.897·37-s − 0.355·39-s + 0.784·41-s + 0.413·43-s + 0.785·45-s − 0.246·47-s − 0.833·49-s − 0.0208·51-s − 1.20·53-s − 1.26·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 13 | \( 1 - T \) |
| 29 | \( 1 - T \) |
good | 3 | \( 1 + 2.22T + 3T^{2} \) |
| 5 | \( 1 - 2.71T + 5T^{2} \) |
| 7 | \( 1 + 1.08T + 7T^{2} \) |
| 11 | \( 1 + 3.45T + 11T^{2} \) |
| 17 | \( 1 - 0.0669T + 17T^{2} \) |
| 19 | \( 1 + 7.68T + 19T^{2} \) |
| 23 | \( 1 - 6.33T + 23T^{2} \) |
| 31 | \( 1 - 6.08T + 31T^{2} \) |
| 37 | \( 1 - 5.46T + 37T^{2} \) |
| 41 | \( 1 - 5.02T + 41T^{2} \) |
| 43 | \( 1 - 2.71T + 43T^{2} \) |
| 47 | \( 1 + 1.68T + 47T^{2} \) |
| 53 | \( 1 + 8.77T + 53T^{2} \) |
| 59 | \( 1 - 3.74T + 59T^{2} \) |
| 61 | \( 1 + 12.8T + 61T^{2} \) |
| 67 | \( 1 - 13.7T + 67T^{2} \) |
| 71 | \( 1 + 3.42T + 71T^{2} \) |
| 73 | \( 1 + 4.09T + 73T^{2} \) |
| 79 | \( 1 + 13.8T + 79T^{2} \) |
| 83 | \( 1 + 12.4T + 83T^{2} \) |
| 89 | \( 1 - 4.56T + 89T^{2} \) |
| 97 | \( 1 - 16.1T + 97T^{2} \) |
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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
−7.61041256611062830140568259728, −6.59190187112755671579611961058, −6.24997280175791695180237966159, −5.73840959324676222945504614279, −4.95198638456231069104583814821, −4.42619511236657899791222494106, −3.00940590733924698394852486958, −2.31744108971742468701819422231, −1.16143971979236232990214745930, 0,
1.16143971979236232990214745930, 2.31744108971742468701819422231, 3.00940590733924698394852486958, 4.42619511236657899791222494106, 4.95198638456231069104583814821, 5.73840959324676222945504614279, 6.24997280175791695180237966159, 6.59190187112755671579611961058, 7.61041256611062830140568259728