| L(s) = 1 | + 8·7-s + 6·11-s − 2·13-s − 6·17-s + 7·19-s + 6·23-s + 5·25-s − 4·31-s − 20·37-s + 9·41-s − 4·43-s + 34·49-s + 6·53-s + 9·59-s + 4·61-s − 7·67-s + 6·71-s + 73-s + 48·77-s − 4·79-s + 6·83-s + 6·89-s − 16·91-s − 17·97-s − 4·103-s + 16·109-s − 30·113-s + ⋯ |
| L(s) = 1 | + 3.02·7-s + 1.80·11-s − 0.554·13-s − 1.45·17-s + 1.60·19-s + 1.25·23-s + 25-s − 0.718·31-s − 3.28·37-s + 1.40·41-s − 0.609·43-s + 34/7·49-s + 0.824·53-s + 1.17·59-s + 0.512·61-s − 0.855·67-s + 0.712·71-s + 0.117·73-s + 5.47·77-s − 0.450·79-s + 0.658·83-s + 0.635·89-s − 1.67·91-s − 1.72·97-s − 0.394·103-s + 1.53·109-s − 2.82·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7485696 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7485696 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.315981290\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.315981290\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | | \( 1 \) |
| 19 | $C_2$ | \( 1 - 7 T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 6 T + 19 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 6 T + 13 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 9 T + 40 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 4 T - 27 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 6 T - 17 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 9 T + 22 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 4 T - 45 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 7 T - 18 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 6 T - 35 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - T - 72 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 6 T - 53 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 17 T + 192 T^{2} + 17 p T^{3} + p^{2} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.866965424272802510991259639094, −8.809842204645031191082320694440, −8.130877166547134200525843133549, −8.087322983214289735111935441254, −7.47048054528617358571003891896, −7.01474542960158354985191035048, −6.81407231652924487367687030366, −6.73228229957901982452071099600, −5.58974977633622392681504014934, −5.44641619484999049912524165011, −5.14055131386955070080165044730, −4.76003577771701243787518705418, −4.25561498081208946276202095250, −4.11220154871372790066642210530, −3.38410425171896998797815321964, −2.90224816794918410657245787754, −1.96306291386956107663605922492, −1.92066422338471273666688719567, −1.28057964645246709468571462560, −0.846566516559406949925808136369,
0.846566516559406949925808136369, 1.28057964645246709468571462560, 1.92066422338471273666688719567, 1.96306291386956107663605922492, 2.90224816794918410657245787754, 3.38410425171896998797815321964, 4.11220154871372790066642210530, 4.25561498081208946276202095250, 4.76003577771701243787518705418, 5.14055131386955070080165044730, 5.44641619484999049912524165011, 5.58974977633622392681504014934, 6.73228229957901982452071099600, 6.81407231652924487367687030366, 7.01474542960158354985191035048, 7.47048054528617358571003891896, 8.087322983214289735111935441254, 8.130877166547134200525843133549, 8.809842204645031191082320694440, 8.866965424272802510991259639094
