L(s) = 1 | − 5·4-s − 10·5-s − 6·9-s + 14·16-s + 50·20-s + 45·25-s + 40·31-s + 30·36-s + 60·45-s − 10·49-s + 8·59-s − 40·64-s − 16·71-s − 140·80-s + 9·81-s − 225·100-s − 200·124-s − 100·125-s + 127-s + 131-s + 137-s + 139-s − 84·144-s + 149-s + 151-s − 400·155-s + 157-s + ⋯ |
L(s) = 1 | − 5/2·4-s − 4.47·5-s − 2·9-s + 7/2·16-s + 11.1·20-s + 9·25-s + 7.18·31-s + 5·36-s + 8.94·45-s − 1.42·49-s + 1.04·59-s − 5·64-s − 1.89·71-s − 15.6·80-s + 81-s − 22.5·100-s − 17.9·124-s − 8.94·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 7·144-s + 0.0819·149-s + 0.0813·151-s − 32.1·155-s + 0.0798·157-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 11^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 11^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.2031092474\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2031092474\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( ( 1 + p T + 3 p T^{2} + p^{2} T^{3} + p^{2} T^{4} )^{2} \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + 5 T^{2} + 11 T^{4} + 25 T^{6} + 61 T^{8} + 25 p^{2} T^{10} + 11 p^{4} T^{12} + 5 p^{6} T^{14} + p^{8} T^{16} \) |
| 3 | \( ( 1 + p T^{2} + p^{2} T^{4} + p^{3} T^{6} + p^{4} T^{8} )^{2} \) |
| 7 | \( 1 + 10 T^{2} + 11 T^{4} + 200 T^{6} + 3781 T^{8} + 200 p^{2} T^{10} + 11 p^{4} T^{12} + 10 p^{6} T^{14} + p^{8} T^{16} \) |
| 13 | \( 1 - 50 T^{2} + 1331 T^{4} - 25000 T^{6} + 363061 T^{8} - 25000 p^{2} T^{10} + 1331 p^{4} T^{12} - 50 p^{6} T^{14} + p^{8} T^{16} \) |
| 17 | \( 1 + 70 T^{2} + 2651 T^{4} + 68600 T^{6} + 1327141 T^{8} + 68600 p^{2} T^{10} + 2651 p^{4} T^{12} + 70 p^{6} T^{14} + p^{8} T^{16} \) |
| 19 | \( ( 1 - p T^{2} + p^{2} T^{4} - p^{3} T^{6} + p^{4} T^{8} )^{2} \) |
| 23 | \( ( 1 - p T^{2} )^{8} \) |
| 29 | \( ( 1 - p T^{2} + p^{2} T^{4} - p^{3} T^{6} + p^{4} T^{8} )^{2} \) |
| 31 | \( ( 1 - 20 T + 209 T^{2} - 1600 T^{3} + 9841 T^{4} - 1600 p T^{5} + 209 p^{2} T^{6} - 20 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 37 | \( ( 1 + p T^{2} + p^{2} T^{4} + p^{3} T^{6} + p^{4} T^{8} )^{2} \) |
| 41 | \( ( 1 - p T^{2} + p^{2} T^{4} - p^{3} T^{6} + p^{4} T^{8} )^{2} \) |
| 43 | \( ( 1 - 3522 T^{4} + p^{4} T^{8} )^{2} \) |
| 47 | \( ( 1 + p T^{2} + p^{2} T^{4} + p^{3} T^{6} + p^{4} T^{8} )^{2} \) |
| 53 | \( ( 1 + p T^{2} + p^{2} T^{4} + p^{3} T^{6} + p^{4} T^{8} )^{2} \) |
| 59 | \( ( 1 - 4 T - 43 T^{2} + 408 T^{3} + 905 T^{4} + 408 p T^{5} - 43 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 - p T^{2} + p^{2} T^{4} - p^{3} T^{6} + p^{4} T^{8} )^{2} \) |
| 67 | \( ( 1 - p T^{2} )^{8} \) |
| 71 | \( ( 1 + 8 T - 7 T^{2} - 624 T^{3} - 4495 T^{4} - 624 p T^{5} - 7 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 73 | \( 1 + 10 T^{2} - 5269 T^{4} + 200 T^{6} + 28505221 T^{8} + 200 p^{2} T^{10} - 5269 p^{4} T^{12} + 10 p^{6} T^{14} + p^{8} T^{16} \) |
| 79 | \( ( 1 - p T^{2} + p^{2} T^{4} - p^{3} T^{6} + p^{4} T^{8} )^{2} \) |
| 83 | \( 1 - 370 T^{2} + 75251 T^{4} - 10130600 T^{6} + 985743541 T^{8} - 10130600 p^{2} T^{10} + 75251 p^{4} T^{12} - 370 p^{6} T^{14} + p^{8} T^{16} \) |
| 89 | \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{4} \) |
| 97 | \( ( 1 + p T^{2} + p^{2} T^{4} + p^{3} T^{6} + p^{4} T^{8} )^{2} \) |
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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
−4.49296070572724849731893375169, −4.44511932053291882400995129062, −4.36051420484588114191276144499, −4.29822570695035416665600697241, −4.06733012926720557251421115441, −4.05651294628167799671206376295, −4.01821926490180502078822441261, −3.81641808073273028376633041448, −3.39088455990360914138046953108, −3.37616625491496910978177356124, −3.24209528170556487090054960087, −3.12475276566318883998019062865, −3.07523112776643168439152179468, −3.00535894091498237184184576529, −2.69139613564784564134322976748, −2.57063137206641762348552611520, −2.43251737404647296561262193985, −2.12455713919972767991096341077, −1.68066974952694402511355172313, −1.52191771568781573467269811470, −1.09324233311305879768541280273, −0.827110009784552620750718314605, −0.74965194608776641336744147747, −0.36568054140630699982563828319, −0.27026797428729992857328065727,
0.27026797428729992857328065727, 0.36568054140630699982563828319, 0.74965194608776641336744147747, 0.827110009784552620750718314605, 1.09324233311305879768541280273, 1.52191771568781573467269811470, 1.68066974952694402511355172313, 2.12455713919972767991096341077, 2.43251737404647296561262193985, 2.57063137206641762348552611520, 2.69139613564784564134322976748, 3.00535894091498237184184576529, 3.07523112776643168439152179468, 3.12475276566318883998019062865, 3.24209528170556487090054960087, 3.37616625491496910978177356124, 3.39088455990360914138046953108, 3.81641808073273028376633041448, 4.01821926490180502078822441261, 4.05651294628167799671206376295, 4.06733012926720557251421115441, 4.29822570695035416665600697241, 4.36051420484588114191276144499, 4.44511932053291882400995129062, 4.49296070572724849731893375169
Plot not available for L-functions of degree greater than 10.