| L(s) = 1 | − 3·7-s + 3·11-s − 13-s − 7·17-s − 2·19-s + 23-s − 6·29-s − 10·31-s − 37-s + 41-s − 4·43-s − 4·47-s + 2·49-s + 53-s + 13·61-s − 8·67-s − 5·71-s − 10·73-s − 9·77-s − 79-s − 6·83-s − 9·89-s + 3·91-s + 97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | − 1.13·7-s + 0.904·11-s − 0.277·13-s − 1.69·17-s − 0.458·19-s + 0.208·23-s − 1.11·29-s − 1.79·31-s − 0.164·37-s + 0.156·41-s − 0.609·43-s − 0.583·47-s + 2/7·49-s + 0.137·53-s + 1.66·61-s − 0.977·67-s − 0.593·71-s − 1.17·73-s − 1.02·77-s − 0.112·79-s − 0.658·83-s − 0.953·89-s + 0.314·91-s + 0.101·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 93600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 93600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 + 7 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 + T + p T^{2} \) |
| 41 | \( 1 - T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 13 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + 5 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 9 T + p T^{2} \) |
| 97 | \( 1 - T + p T^{2} \) |
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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
−14.40828259835849, −13.69874377049013, −13.18997180984091, −12.91214095954482, −12.55754076397329, −11.73450210126054, −11.43599717770412, −10.88704345272466, −10.39021676134074, −9.717797849398318, −9.353323847318580, −8.879859990860667, −8.562899408581669, −7.658104123666399, −7.104185429364936, −6.692477011582633, −6.347516616049421, −5.634721257760045, −5.142261572291142, −4.205476722103716, −4.011589544816043, −3.319670933027645, −2.659135944593553, −1.979724148237976, −1.371764018724948, 0, 0,
1.371764018724948, 1.979724148237976, 2.659135944593553, 3.319670933027645, 4.011589544816043, 4.205476722103716, 5.142261572291142, 5.634721257760045, 6.347516616049421, 6.692477011582633, 7.104185429364936, 7.658104123666399, 8.562899408581669, 8.879859990860667, 9.353323847318580, 9.717797849398318, 10.39021676134074, 10.88704345272466, 11.43599717770412, 11.73450210126054, 12.55754076397329, 12.91214095954482, 13.18997180984091, 13.69874377049013, 14.40828259835849
