| L(s) = 1 | − 4.78e3·5-s − 1.96e4·9-s + 2.48e5·13-s + 2.55e6·17-s − 2.33e6·25-s + 3.16e7·29-s + 2.33e8·37-s + 2.64e8·41-s + 9.42e7·45-s + 5.20e8·49-s + 9.78e8·53-s + 1.10e9·61-s − 1.19e9·65-s − 6.17e9·73-s + 3.87e8·81-s − 1.22e10·85-s − 1.61e9·89-s − 1.64e10·97-s − 8.37e9·101-s + 1.46e10·109-s + 7.13e10·113-s − 4.89e9·117-s + 3.94e10·121-s + 8.53e10·125-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | − 1.53·5-s − 1/3·9-s + 0.669·13-s + 1.79·17-s − 0.239·25-s + 1.54·29-s + 3.36·37-s + 2.28·41-s + 0.510·45-s + 1.84·49-s + 2.34·53-s + 1.30·61-s − 1.02·65-s − 2.97·73-s + 1/9·81-s − 2.75·85-s − 0.289·89-s − 1.92·97-s − 0.796·101-s + 0.951·109-s + 3.87·113-s − 0.223·117-s + 1.52·121-s + 2.79·125-s − 2.36·145-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+5)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{11}{2})\) |
\(\approx\) |
\(2.641105991\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.641105991\) |
| \(L(6)\) |
|
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 | $C_2$ | \( 1 + p^{9} T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 + 2394 T + p^{10} T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 - 10628690 p^{2} T^{2} + p^{20} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 39459030530 T^{2} + p^{20} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 9562 p T + p^{10} T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 1276794 T + p^{10} T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 11523143800034 T^{2} + p^{20} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 81559494158498 T^{2} + p^{20} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 15807870 T + p^{10} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 684980179236910 T^{2} + p^{20} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 116655866 T + p^{10} T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 132356106 T + p^{10} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 42342726693304706 T^{2} + p^{20} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 81876278110171010 T^{2} + p^{20} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 489324942 T + p^{10} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 788771550043537730 T^{2} + p^{20} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 551229530 T + p^{10} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 3601617710717704226 T^{2} + p^{20} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 398656133613437470 T^{2} + p^{20} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 3085329518 T + p^{10} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 18836980136966441234 T^{2} + p^{20} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 132318476153326 p^{2} T^{2} + p^{20} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 806913486 T + p^{10} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 85044638 p T + p^{10} T^{2} )^{2} \) |
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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
−13.73302934216819499252181617970, −13.21929605873588725002562627450, −12.23553635224157496155883521429, −12.18158389597844463664844946581, −11.32882523839329218335256018034, −11.16715365520268178031908473086, −10.12381208289513085802672841531, −9.730211222383538795833827224356, −8.675540056306968511003528786503, −8.267073504673874150014378238119, −7.55039276004413754811829421960, −7.30565954934165250335483548338, −5.85404838861916723794994663911, −5.83779123883587896504595365917, −4.28289735737920386207126632934, −4.14724702459550560409899274774, −3.17503690142517348871708932271, −2.47383674031320903336843861426, −0.864566451518598459976327252506, −0.76131703395462548031750471792,
0.76131703395462548031750471792, 0.864566451518598459976327252506, 2.47383674031320903336843861426, 3.17503690142517348871708932271, 4.14724702459550560409899274774, 4.28289735737920386207126632934, 5.83779123883587896504595365917, 5.85404838861916723794994663911, 7.30565954934165250335483548338, 7.55039276004413754811829421960, 8.267073504673874150014378238119, 8.675540056306968511003528786503, 9.730211222383538795833827224356, 10.12381208289513085802672841531, 11.16715365520268178031908473086, 11.32882523839329218335256018034, 12.18158389597844463664844946581, 12.23553635224157496155883521429, 13.21929605873588725002562627450, 13.73302934216819499252181617970
