| L(s) = 1 | − 2·3-s + 2·7-s + 3·9-s + 2·11-s − 3·17-s + 5·19-s − 4·21-s − 3·23-s − 4·27-s + 8·29-s + 13·31-s − 4·33-s − 4·37-s + 2·43-s − 7·47-s − 6·49-s + 6·51-s − 53-s − 10·57-s + 18·59-s + 9·61-s + 6·63-s + 18·67-s + 6·69-s + 20·71-s + 16·73-s + 4·77-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 0.755·7-s + 9-s + 0.603·11-s − 0.727·17-s + 1.14·19-s − 0.872·21-s − 0.625·23-s − 0.769·27-s + 1.48·29-s + 2.33·31-s − 0.696·33-s − 0.657·37-s + 0.304·43-s − 1.02·47-s − 6/7·49-s + 0.840·51-s − 0.137·53-s − 1.32·57-s + 2.34·59-s + 1.15·61-s + 0.755·63-s + 2.19·67-s + 0.722·69-s + 2.37·71-s + 1.87·73-s + 0.455·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2250000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2250000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.117570029\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.117570029\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 + T )^{2} \) |
| 5 | | \( 1 \) |
| good | 7 | $D_{4}$ | \( 1 - 2 T + 10 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 2 T + 18 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 17 | $D_{4}$ | \( 1 + 3 T + 35 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 5 T + 33 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 3 T + 37 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_4$ | \( 1 - 8 T + 54 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 13 T + 103 T^{2} - 13 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 2 T + 42 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 7 T + 75 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + T + 75 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 18 T + 194 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 9 T + 41 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 18 T + 170 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 20 T + 222 T^{2} - 20 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 16 T + 190 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 21 T + 257 T^{2} - 21 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 3 T + 107 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 10 T + 198 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
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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
−9.726432859737664682156228231807, −9.497005641017377038809873092866, −8.824577996080134723067605253424, −8.358961407170585384569609393986, −8.075526437938737313471231979071, −7.83928116943265167313604277616, −7.03070183816715098003664818653, −6.75246493362657133273706625553, −6.42152103991828550703463970755, −6.20275403311905355189319897979, −5.38905574574784813019059822265, −5.10094224524493417130218281167, −4.79421320658196128892458039315, −4.42362928754013325407308539896, −3.59172372259001658629139689784, −3.51687206381026694744303989327, −2.28290295872167970424742662503, −2.18517134564642826062018053617, −0.978694031598392741177382144865, −0.849676501480681629201491159721,
0.849676501480681629201491159721, 0.978694031598392741177382144865, 2.18517134564642826062018053617, 2.28290295872167970424742662503, 3.51687206381026694744303989327, 3.59172372259001658629139689784, 4.42362928754013325407308539896, 4.79421320658196128892458039315, 5.10094224524493417130218281167, 5.38905574574784813019059822265, 6.20275403311905355189319897979, 6.42152103991828550703463970755, 6.75246493362657133273706625553, 7.03070183816715098003664818653, 7.83928116943265167313604277616, 8.075526437938737313471231979071, 8.358961407170585384569609393986, 8.824577996080134723067605253424, 9.497005641017377038809873092866, 9.726432859737664682156228231807
