| L(s) = 1 | + 2.59·5-s + 7-s + 4.51·11-s + 13-s + 0.945·17-s + 4.05·19-s − 0.273·23-s + 1.72·25-s − 2.46·29-s − 2.32·31-s + 2.59·35-s + 1.78·37-s + 6.40·41-s + 10.4·43-s − 12.1·47-s + 49-s + 6.27·53-s + 11.7·55-s − 2.72·59-s − 2.27·61-s + 2.59·65-s + 15.8·67-s + 3.27·71-s − 1.50·73-s + 4.51·77-s − 14.7·79-s − 0.945·83-s + ⋯ |
| L(s) = 1 | + 1.15·5-s + 0.377·7-s + 1.36·11-s + 0.277·13-s + 0.229·17-s + 0.930·19-s − 0.0569·23-s + 0.345·25-s − 0.456·29-s − 0.418·31-s + 0.438·35-s + 0.292·37-s + 1.00·41-s + 1.59·43-s − 1.77·47-s + 0.142·49-s + 0.861·53-s + 1.57·55-s − 0.354·59-s − 0.291·61-s + 0.321·65-s + 1.93·67-s + 0.388·71-s − 0.176·73-s + 0.514·77-s − 1.65·79-s − 0.103·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9072 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9072 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.469175724\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.469175724\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 5 | \( 1 - 2.59T + 5T^{2} \) |
| 11 | \( 1 - 4.51T + 11T^{2} \) |
| 13 | \( 1 - T + 13T^{2} \) |
| 17 | \( 1 - 0.945T + 17T^{2} \) |
| 19 | \( 1 - 4.05T + 19T^{2} \) |
| 23 | \( 1 + 0.273T + 23T^{2} \) |
| 29 | \( 1 + 2.46T + 29T^{2} \) |
| 31 | \( 1 + 2.32T + 31T^{2} \) |
| 37 | \( 1 - 1.78T + 37T^{2} \) |
| 41 | \( 1 - 6.40T + 41T^{2} \) |
| 43 | \( 1 - 10.4T + 43T^{2} \) |
| 47 | \( 1 + 12.1T + 47T^{2} \) |
| 53 | \( 1 - 6.27T + 53T^{2} \) |
| 59 | \( 1 + 2.72T + 59T^{2} \) |
| 61 | \( 1 + 2.27T + 61T^{2} \) |
| 67 | \( 1 - 15.8T + 67T^{2} \) |
| 71 | \( 1 - 3.27T + 71T^{2} \) |
| 73 | \( 1 + 1.50T + 73T^{2} \) |
| 79 | \( 1 + 14.7T + 79T^{2} \) |
| 83 | \( 1 + 0.945T + 83T^{2} \) |
| 89 | \( 1 - 14.3T + 89T^{2} \) |
| 97 | \( 1 + 11.4T + 97T^{2} \) |
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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
−7.66895622605028548101330604521, −7.00062024358702394383049678718, −6.23286477164653265634284251459, −5.77256933548015013723448711351, −5.09842492159626030127868293790, −4.18702204619048661703442614580, −3.50676426229852747374846894089, −2.49292613774783915446757985674, −1.66129338501147757300177426639, −0.988222803437643961103715042710,
0.988222803437643961103715042710, 1.66129338501147757300177426639, 2.49292613774783915446757985674, 3.50676426229852747374846894089, 4.18702204619048661703442614580, 5.09842492159626030127868293790, 5.77256933548015013723448711351, 6.23286477164653265634284251459, 7.00062024358702394383049678718, 7.66895622605028548101330604521
