| L(s) = 1 | + 40·9-s + 92·25-s − 176·41-s − 104·49-s + 676·81-s − 1.16e3·89-s − 200·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 968·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 3.68e3·225-s + ⋯ |
| L(s) = 1 | + 40/9·9-s + 3.67·25-s − 4.29·41-s − 2.12·49-s + 8.34·81-s − 13.1·89-s − 1.65·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s − 5.72·169-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + 0.00518·193-s + 0.00507·197-s + 0.00502·199-s + 0.00473·211-s + 0.00448·223-s + 16.3·225-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{64} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{64} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.4338747003\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4338747003\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( ( 1 - 46 T^{2} + p^{4} T^{4} )^{2} \) |
| good | 3 | \( ( 1 - 10 T^{2} + p^{4} T^{4} )^{4} \) |
| 7 | \( ( 1 + 26 T^{2} + p^{4} T^{4} )^{4} \) |
| 11 | \( ( 1 + 50 T^{2} + p^{4} T^{4} )^{4} \) |
| 13 | \( ( 1 + 242 T^{2} + p^{4} T^{4} )^{4} \) |
| 17 | \( ( 1 - 194 T^{2} + p^{4} T^{4} )^{4} \) |
| 19 | \( ( 1 + 530 T^{2} + p^{4} T^{4} )^{4} \) |
| 23 | \( ( 1 + 410 T^{2} + p^{4} T^{4} )^{4} \) |
| 29 | \( ( 1 - 1198 T^{2} + p^{4} T^{4} )^{4} \) |
| 31 | \( ( 1 + 1150 T^{2} + p^{4} T^{4} )^{4} \) |
| 37 | \( ( 1 + 338 T^{2} + p^{4} T^{4} )^{4} \) |
| 41 | \( ( 1 + 22 T + p^{2} T^{2} )^{8} \) |
| 43 | \( ( 1 - 170 T^{2} + p^{4} T^{4} )^{4} \) |
| 47 | \( ( 1 + 4346 T^{2} + p^{4} T^{4} )^{4} \) |
| 53 | \( ( 1 + 4754 T^{2} + p^{4} T^{4} )^{4} \) |
| 59 | \( ( 1 + 6770 T^{2} + p^{4} T^{4} )^{4} \) |
| 61 | \( ( 1 - 5326 T^{2} + p^{4} T^{4} )^{4} \) |
| 67 | \( ( 1 - 5450 T^{2} + p^{4} T^{4} )^{4} \) |
| 71 | \( ( 1 - 9314 T^{2} + p^{4} T^{4} )^{4} \) |
| 73 | \( ( 1 - 4514 T^{2} + p^{4} T^{4} )^{4} \) |
| 79 | \( ( 1 - p T )^{8}( 1 + p T )^{8} \) |
| 83 | \( ( 1 - 7946 T^{2} + p^{4} T^{4} )^{4} \) |
| 89 | \( ( 1 + 146 T + p^{2} T^{2} )^{8} \) |
| 97 | \( ( 1 - 15362 T^{2} + p^{4} T^{4} )^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.87155183601534757909654895512, −3.80315076817805001793510883276, −3.72304865709449217570900730852, −3.60825974129404079908963242275, −3.30348922753349584337643748588, −3.15057287062650379460896982196, −3.10981992344887429938853796903, −3.03327583626180496754828739314, −2.97767492948325278309489420319, −2.77971990274096894626213068468, −2.39135980402347398176459310606, −2.39043237721998554962098679161, −2.32044046092389322388733498034, −2.17426145890645585376385509350, −1.77626216620086984927337290820, −1.62543881496319826818394941533, −1.48280977423416698951450081004, −1.37816848971554490851505166497, −1.31518507967633625688422412295, −1.23029604134116961306875750339, −1.18638932925301745769951804802, −1.09334268446599956154775636220, −0.44756476308975170954149282885, −0.31408021535444123282075995387, −0.04433267598712282554177272675,
0.04433267598712282554177272675, 0.31408021535444123282075995387, 0.44756476308975170954149282885, 1.09334268446599956154775636220, 1.18638932925301745769951804802, 1.23029604134116961306875750339, 1.31518507967633625688422412295, 1.37816848971554490851505166497, 1.48280977423416698951450081004, 1.62543881496319826818394941533, 1.77626216620086984927337290820, 2.17426145890645585376385509350, 2.32044046092389322388733498034, 2.39043237721998554962098679161, 2.39135980402347398176459310606, 2.77971990274096894626213068468, 2.97767492948325278309489420319, 3.03327583626180496754828739314, 3.10981992344887429938853796903, 3.15057287062650379460896982196, 3.30348922753349584337643748588, 3.60825974129404079908963242275, 3.72304865709449217570900730852, 3.80315076817805001793510883276, 3.87155183601534757909654895512
Plot not available for L-functions of degree greater than 10.
