| L(s) = 1 | + 2·5-s − 17·9-s − 96·13-s − 2·17-s + 51·25-s − 96·29-s − 98·37-s − 192·41-s − 34·45-s − 98·49-s − 50·53-s − 2·61-s − 192·65-s − 190·73-s + 81·81-s − 4·85-s − 190·89-s + 576·97-s + 146·101-s − 142·109-s + 384·113-s + 1.63e3·117-s + 47·121-s + 198·125-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | + 2/5·5-s − 1.88·9-s − 7.38·13-s − 0.117·17-s + 2.03·25-s − 3.31·29-s − 2.64·37-s − 4.68·41-s − 0.755·45-s − 2·49-s − 0.943·53-s − 0.0327·61-s − 2.95·65-s − 2.60·73-s + 81-s − 0.0470·85-s − 2.13·89-s + 5.93·97-s + 1.44·101-s − 1.30·109-s + 3.39·113-s + 13.9·117-s + 0.388·121-s + 1.58·125-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.01365970572\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.01365970572\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| 7 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| good | 3 | $C_2^3$ | \( 1 + 17 T^{2} + 208 T^{4} + 17 p^{4} T^{6} + p^{8} T^{8} \) |
| 5 | $C_2^2$ | \( ( 1 - T - 24 T^{2} - p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 11 | $C_2^3$ | \( 1 - 47 T^{2} - 12432 T^{4} - 47 p^{4} T^{6} + p^{8} T^{8} \) |
| 13 | $C_2$ | \( ( 1 + 24 T + p^{2} T^{2} )^{4} \) |
| 17 | $C_2^2$ | \( ( 1 + T - 288 T^{2} + p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 19 | $C_2^3$ | \( 1 + 673 T^{2} + 322608 T^{4} + 673 p^{4} T^{6} + p^{8} T^{8} \) |
| 23 | $C_2^3$ | \( 1 + 1009 T^{2} + 738240 T^{4} + 1009 p^{4} T^{6} + p^{8} T^{8} \) |
| 29 | $C_2$ | \( ( 1 + 24 T + p^{2} T^{2} )^{4} \) |
| 31 | $C_2^3$ | \( 1 + 241 T^{2} - 865440 T^{4} + 241 p^{4} T^{6} + p^{8} T^{8} \) |
| 37 | $C_2^2$ | \( ( 1 + 49 T + 1032 T^{2} + 49 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 48 T + p^{2} T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 - 3122 T^{2} + p^{4} T^{4} )^{2} \) |
| 47 | $C_2^3$ | \( 1 + 1393 T^{2} - 2939232 T^{4} + 1393 p^{4} T^{6} + p^{8} T^{8} \) |
| 53 | $C_2^2$ | \( ( 1 + 25 T - 2184 T^{2} + 25 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 59 | $C_2^3$ | \( 1 + 6673 T^{2} + 32411568 T^{4} + 6673 p^{4} T^{6} + p^{8} T^{8} \) |
| 61 | $C_2^2$ | \( ( 1 + T - 3720 T^{2} + p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 67 | $C_2^3$ | \( 1 + 4753 T^{2} + 2439888 T^{4} + 4753 p^{4} T^{6} + p^{8} T^{8} \) |
| 71 | $C_2^2$ | \( ( 1 - 866 T^{2} + p^{4} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 + 95 T + 3696 T^{2} + 95 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 79 | $C_2^3$ | \( 1 + 10801 T^{2} + 77711520 T^{4} + 10801 p^{4} T^{6} + p^{8} T^{8} \) |
| 83 | $C_2^2$ | \( ( 1 - 8594 T^{2} + p^{4} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 + 95 T + 1104 T^{2} + 95 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 144 T + p^{2} T^{2} )^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.62696430732388514583346274443, −7.54949489544941922417424132344, −7.29778855300509317129268576882, −7.02564441896097269044328683695, −7.01427346443869412904306104481, −6.73740712523444394382001175166, −6.50170782682663882379945171289, −5.85746225703379711626919228244, −5.62491079840176051253075442472, −5.46224197537040400124681335398, −5.21411894544555510434594442273, −4.85657467050835926375502413978, −4.78685603516214480234950170554, −4.77252895953908822375475535372, −4.55020924107429380398225411575, −3.62207386917506454480634868121, −3.18459879211299716230758214109, −3.09491007957084283877198376297, −3.09322934258198497851951944187, −2.48510538827698115864032007646, −1.97773768532242015162113459173, −1.95419804671742730695352615418, −1.91334705527746563144044494219, −0.25845399011717106772599613772, −0.06667605047837658086524589844,
0.06667605047837658086524589844, 0.25845399011717106772599613772, 1.91334705527746563144044494219, 1.95419804671742730695352615418, 1.97773768532242015162113459173, 2.48510538827698115864032007646, 3.09322934258198497851951944187, 3.09491007957084283877198376297, 3.18459879211299716230758214109, 3.62207386917506454480634868121, 4.55020924107429380398225411575, 4.77252895953908822375475535372, 4.78685603516214480234950170554, 4.85657467050835926375502413978, 5.21411894544555510434594442273, 5.46224197537040400124681335398, 5.62491079840176051253075442472, 5.85746225703379711626919228244, 6.50170782682663882379945171289, 6.73740712523444394382001175166, 7.01427346443869412904306104481, 7.02564441896097269044328683695, 7.29778855300509317129268576882, 7.54949489544941922417424132344, 7.62696430732388514583346274443
