L(s) = 1 | + (0.707 − 1.22i)3-s + (1.41 + 2.44i)5-s + (0.500 + 0.866i)9-s + (−3 + 5.19i)11-s + 5.65·13-s + 4·15-s + (0.707 − 1.22i)17-s + (−2.12 − 3.67i)19-s + (−2 − 3.46i)23-s + (−1.49 + 2.59i)25-s + 5.65·27-s − 6·29-s + (1.41 − 2.44i)31-s + (4.24 + 7.34i)33-s + (−1 − 1.73i)37-s + ⋯ |
L(s) = 1 | + (0.408 − 0.707i)3-s + (0.632 + 1.09i)5-s + (0.166 + 0.288i)9-s + (−0.904 + 1.56i)11-s + 1.56·13-s + 1.03·15-s + (0.171 − 0.297i)17-s + (−0.486 − 0.842i)19-s + (−0.417 − 0.722i)23-s + (−0.299 + 0.519i)25-s + 1.08·27-s − 1.11·29-s + (0.254 − 0.439i)31-s + (0.738 + 1.27i)33-s + (−0.164 − 0.284i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.947 - 0.318i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.947 - 0.318i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.70385 + 0.278659i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.70385 + 0.278659i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + (-0.707 + 1.22i)T + (-1.5 - 2.59i)T^{2} \) |
| 5 | \( 1 + (-1.41 - 2.44i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (3 - 5.19i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 - 5.65T + 13T^{2} \) |
| 17 | \( 1 + (-0.707 + 1.22i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (2.12 + 3.67i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (2 + 3.46i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 + (-1.41 + 2.44i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (1 + 1.73i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 - 1.41T + 41T^{2} \) |
| 43 | \( 1 - 10T + 43T^{2} \) |
| 47 | \( 1 + (1.41 + 2.44i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-1 + 1.73i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-0.707 + 1.22i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (4.24 + 7.34i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (2 - 3.46i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 12T + 71T^{2} \) |
| 73 | \( 1 + (4.94 - 8.57i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-2 - 3.46i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 1.41T + 83T^{2} \) |
| 89 | \( 1 + (2.12 + 3.67i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 12.7T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.09671417251668089470172585457, −10.52379048533442829333039670175, −9.651313956686941468256884325898, −8.445950664487897905451817883930, −7.43478124881347784855257318587, −6.83772341731112096149200972624, −5.79851986796422953478987648181, −4.36992302922154361127194932470, −2.73335477329171616733513866740, −1.93243160170244369798001277600,
1.29902002544199751591309284698, 3.25999778672254357411104497079, 4.17165938053900576729287966017, 5.59035854755148429046805301169, 6.05676351082588739424628170975, 7.933537349227914345196843327074, 8.720646527583337594710292401916, 9.221887687509824836371587989742, 10.34786362404510305668672771919, 10.99840720428763993783118710649