L(s) = 1 | + 4.93·7-s − 2.41·11-s + 2.90·13-s + 6.86·17-s − 4.17·19-s + 3.35·23-s + 5.19·29-s − 6.17·31-s + 7.84·37-s − 5.87·41-s + 4.93·43-s − 11.9·47-s + 17.3·49-s + 8.54·53-s − 1.05·59-s + 9.17·61-s + 4.05·67-s − 14.1·71-s + 2.02·73-s − 11.9·77-s + 6·79-s − 5.18·83-s + 3.09·89-s + 14.3·91-s + 0.882·97-s + 5.87·101-s + 9.87·103-s + ⋯ |
L(s) = 1 | + 1.86·7-s − 0.727·11-s + 0.806·13-s + 1.66·17-s − 0.958·19-s + 0.700·23-s + 0.964·29-s − 1.10·31-s + 1.28·37-s − 0.917·41-s + 0.752·43-s − 1.73·47-s + 2.47·49-s + 1.17·53-s − 0.136·59-s + 1.17·61-s + 0.495·67-s − 1.68·71-s + 0.237·73-s − 1.35·77-s + 0.675·79-s − 0.569·83-s + 0.327·89-s + 1.50·91-s + 0.0896·97-s + 0.584·101-s + 0.972·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.938531315\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.938531315\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 7 | \( 1 - 4.93T + 7T^{2} \) |
| 11 | \( 1 + 2.41T + 11T^{2} \) |
| 13 | \( 1 - 2.90T + 13T^{2} \) |
| 17 | \( 1 - 6.86T + 17T^{2} \) |
| 19 | \( 1 + 4.17T + 19T^{2} \) |
| 23 | \( 1 - 3.35T + 23T^{2} \) |
| 29 | \( 1 - 5.19T + 29T^{2} \) |
| 31 | \( 1 + 6.17T + 31T^{2} \) |
| 37 | \( 1 - 7.84T + 37T^{2} \) |
| 41 | \( 1 + 5.87T + 41T^{2} \) |
| 43 | \( 1 - 4.93T + 43T^{2} \) |
| 47 | \( 1 + 11.9T + 47T^{2} \) |
| 53 | \( 1 - 8.54T + 53T^{2} \) |
| 59 | \( 1 + 1.05T + 59T^{2} \) |
| 61 | \( 1 - 9.17T + 61T^{2} \) |
| 67 | \( 1 - 4.05T + 67T^{2} \) |
| 71 | \( 1 + 14.1T + 71T^{2} \) |
| 73 | \( 1 - 2.02T + 73T^{2} \) |
| 79 | \( 1 - 6T + 79T^{2} \) |
| 83 | \( 1 + 5.18T + 83T^{2} \) |
| 89 | \( 1 - 3.09T + 89T^{2} \) |
| 97 | \( 1 - 0.882T + 97T^{2} \) |
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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
−7.85517725422828015608953734490, −7.39025885270546363215770114635, −6.39328961432200343118390528182, −5.54722392603342111847967998040, −5.10422821707207809863125267886, −4.39017821318504896659179192781, −3.55899538261094852636255605374, −2.57987890894357134364281087091, −1.67097452986605681025765262254, −0.919436241233365674612642531559,
0.919436241233365674612642531559, 1.67097452986605681025765262254, 2.57987890894357134364281087091, 3.55899538261094852636255605374, 4.39017821318504896659179192781, 5.10422821707207809863125267886, 5.54722392603342111847967998040, 6.39328961432200343118390528182, 7.39025885270546363215770114635, 7.85517725422828015608953734490