| L(s) = 1 | + 5·4-s − 2·9-s − 12·11-s + 12·16-s + 10·29-s + 8·31-s − 10·36-s − 42·41-s − 60·44-s + 20·49-s + 20·59-s + 8·61-s + 15·64-s + 18·71-s + 50·79-s + 3·81-s + 24·99-s − 42·101-s + 30·109-s + 50·116-s + 46·121-s + 40·124-s + 127-s + 131-s + 137-s + 139-s − 24·144-s + ⋯ |
| L(s) = 1 | + 5/2·4-s − 2/3·9-s − 3.61·11-s + 3·16-s + 1.85·29-s + 1.43·31-s − 5/3·36-s − 6.55·41-s − 9.04·44-s + 20/7·49-s + 2.60·59-s + 1.02·61-s + 15/8·64-s + 2.13·71-s + 5.62·79-s + 1/3·81-s + 2.41·99-s − 4.17·101-s + 2.87·109-s + 4.64·116-s + 4.18·121-s + 3.59·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 2·144-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.771524625\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.771524625\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 5 | | \( 1 \) |
| good | 2 | $D_4\times C_2$ | \( 1 - 5 T^{2} + 13 T^{4} - 5 p^{2} T^{6} + p^{4} T^{8} \) |
| 7 | $C_2^2$ | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{4} \) |
| 13 | $C_2^2$ | \( ( 1 - 25 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 - 50 T^{2} + 1123 T^{4} - 50 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $C_2^2$ | \( ( 1 - 7 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 5 T^{2} + 853 T^{4} - 5 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 - 5 T + 63 T^{2} - 5 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $D_{4}$ | \( ( 1 - 4 T + 21 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2}( 1 + 12 T + p T^{2} )^{2} \) |
| 41 | $C_4$ | \( ( 1 + 21 T + 191 T^{2} + 21 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 25 T^{2} + 3493 T^{4} - 25 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 80 T^{2} + 5038 T^{4} - 80 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 40 T^{2} + 4398 T^{4} - 40 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 - 10 T + 63 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 - 4 T + 81 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 10 T^{2} + 7003 T^{4} - 10 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 - 9 T + 131 T^{2} - 9 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 40 T^{2} - 3522 T^{4} - 40 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 - 25 T + 303 T^{2} - 25 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 85 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 + 53 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 365 T^{2} + 52113 T^{4} - 365 p^{2} T^{6} + p^{4} T^{8} \) |
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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
−6.75517523272380675154696923870, −6.43944918541099985455741588931, −6.37132880279268762045785523158, −5.78678932456484747606989159406, −5.74569689445199836623490729166, −5.55092566155764734104371721170, −5.17062265146406838790021623230, −5.11772116988684030709242722812, −5.09457328755616601330751222291, −4.82297160402479758201964185540, −4.61971934042264101553154085310, −3.98773615507736036836740811364, −3.74930003170665451640786100248, −3.63878777804347991647917846809, −3.23772062003150715193663704871, −3.11247991479339906016379191630, −2.75666796808375994024581170538, −2.55312829957554503856751277856, −2.48214053294405845939642355522, −2.27319523376607697393281736405, −1.91916599832901717469399012794, −1.83194035763247316576318712521, −1.19020297183166260820253643671, −0.67312396664317958234246405237, −0.41977842533846591014964943112,
0.41977842533846591014964943112, 0.67312396664317958234246405237, 1.19020297183166260820253643671, 1.83194035763247316576318712521, 1.91916599832901717469399012794, 2.27319523376607697393281736405, 2.48214053294405845939642355522, 2.55312829957554503856751277856, 2.75666796808375994024581170538, 3.11247991479339906016379191630, 3.23772062003150715193663704871, 3.63878777804347991647917846809, 3.74930003170665451640786100248, 3.98773615507736036836740811364, 4.61971934042264101553154085310, 4.82297160402479758201964185540, 5.09457328755616601330751222291, 5.11772116988684030709242722812, 5.17062265146406838790021623230, 5.55092566155764734104371721170, 5.74569689445199836623490729166, 5.78678932456484747606989159406, 6.37132880279268762045785523158, 6.43944918541099985455741588931, 6.75517523272380675154696923870
