| L(s) = 1 | − 9·9-s + 22·11-s − 64·16-s − 250·19-s + 300·29-s + 584·31-s + 994·41-s + 245·49-s − 470·59-s + 964·61-s + 1.17e3·71-s + 2.09e3·79-s + 81·81-s + 1.54e3·89-s − 198·99-s + 1.65e3·101-s − 540·109-s + 363·121-s + 127-s + 131-s + 137-s + 139-s + 576·144-s + 149-s + 151-s + 157-s + 163-s + ⋯ |
| L(s) = 1 | − 1/3·9-s + 0.603·11-s − 16-s − 3.01·19-s + 1.92·29-s + 3.38·31-s + 3.78·41-s + 5/7·49-s − 1.03·59-s + 2.02·61-s + 1.96·71-s + 2.97·79-s + 1/9·81-s + 1.83·89-s − 0.201·99-s + 1.62·101-s − 0.474·109-s + 3/11·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 1/3·144-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 680625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 680625 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.460031458\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.460031458\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_2$ | \( 1 + p^{2} T^{2} \) |
| 5 | | \( 1 \) |
| 11 | $C_1$ | \( ( 1 - p T )^{2} \) |
| good | 2 | $C_2$ | \( ( 1 - p^{2} T + p^{3} T^{2} )( 1 + p^{2} T + p^{3} T^{2} ) \) |
| 7 | $C_2^2$ | \( 1 - 5 p^{2} T^{2} + p^{6} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 230 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 9385 T^{2} + p^{6} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 125 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 5565 T^{2} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 150 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 292 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 20495 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 497 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 115750 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 71485 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 3990 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 235 T + p^{3} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 482 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 62770 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 587 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 509710 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 1045 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 773910 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 770 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 549335 T^{2} + p^{6} T^{4} \) |
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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
−10.10175704855891524106993499010, −9.598312528393085659961943052962, −9.040568237592484271264621987660, −8.887384871684806299025557208748, −8.261142196776996641071910444847, −8.117670677337713457120576166699, −7.61922300554041846726967311750, −6.66360996620280198223327980651, −6.58758795558440740322585414400, −6.29370133277116369067618480684, −5.91318184475124063355783232224, −4.83903957345246434624335780789, −4.73248956341813860261875205944, −4.15074063102958659872289194961, −3.89257729032422102491357137513, −2.83738008635176249434543481530, −2.36819922164309547252848524265, −2.18811264767056741098704001441, −0.77658923422441761457407763300, −0.73215969055530168466454562572,
0.73215969055530168466454562572, 0.77658923422441761457407763300, 2.18811264767056741098704001441, 2.36819922164309547252848524265, 2.83738008635176249434543481530, 3.89257729032422102491357137513, 4.15074063102958659872289194961, 4.73248956341813860261875205944, 4.83903957345246434624335780789, 5.91318184475124063355783232224, 6.29370133277116369067618480684, 6.58758795558440740322585414400, 6.66360996620280198223327980651, 7.61922300554041846726967311750, 8.117670677337713457120576166699, 8.261142196776996641071910444847, 8.887384871684806299025557208748, 9.040568237592484271264621987660, 9.598312528393085659961943052962, 10.10175704855891524106993499010
