L(s) = 1 | + (1.37 − 0.337i)2-s − 3.10·3-s + (1.77 − 0.926i)4-s + (0.0152 + 2.23i)5-s + (−4.26 + 1.04i)6-s + i·7-s + (2.12 − 1.86i)8-s + 6.63·9-s + (0.774 + 3.06i)10-s + 5.18i·11-s + (−5.50 + 2.87i)12-s + 3.37·13-s + (0.337 + 1.37i)14-s + (−0.0472 − 6.94i)15-s + (2.28 − 3.28i)16-s + 4.40i·17-s + ⋯ |
L(s) = 1 | + (0.971 − 0.238i)2-s − 1.79·3-s + (0.886 − 0.463i)4-s + (0.00681 + 0.999i)5-s + (−1.74 + 0.427i)6-s + 0.377i·7-s + (0.750 − 0.661i)8-s + 2.21·9-s + (0.245 + 0.969i)10-s + 1.56i·11-s + (−1.58 + 0.829i)12-s + 0.936·13-s + (0.0901 + 0.367i)14-s + (−0.0122 − 1.79i)15-s + (0.571 − 0.820i)16-s + 1.06i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.745 - 0.666i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.745 - 0.666i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.26896 + 0.484209i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.26896 + 0.484209i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.37 + 0.337i)T \) |
| 5 | \( 1 + (-0.0152 - 2.23i)T \) |
| 7 | \( 1 - iT \) |
good | 3 | \( 1 + 3.10T + 3T^{2} \) |
| 11 | \( 1 - 5.18iT - 11T^{2} \) |
| 13 | \( 1 - 3.37T + 13T^{2} \) |
| 17 | \( 1 - 4.40iT - 17T^{2} \) |
| 19 | \( 1 + 3.64iT - 19T^{2} \) |
| 23 | \( 1 - 2.34iT - 23T^{2} \) |
| 29 | \( 1 + 2.80iT - 29T^{2} \) |
| 31 | \( 1 + 0.829T + 31T^{2} \) |
| 37 | \( 1 + 7.31T + 37T^{2} \) |
| 41 | \( 1 - 4.72T + 41T^{2} \) |
| 43 | \( 1 - 4.34T + 43T^{2} \) |
| 47 | \( 1 + 3.33iT - 47T^{2} \) |
| 53 | \( 1 + 0.861T + 53T^{2} \) |
| 59 | \( 1 + 8.70iT - 59T^{2} \) |
| 61 | \( 1 + 3.83iT - 61T^{2} \) |
| 67 | \( 1 - 6.66T + 67T^{2} \) |
| 71 | \( 1 + 13.5T + 71T^{2} \) |
| 73 | \( 1 + 2.29iT - 73T^{2} \) |
| 79 | \( 1 - 7.50T + 79T^{2} \) |
| 83 | \( 1 - 10.3T + 83T^{2} \) |
| 89 | \( 1 - 7.26T + 89T^{2} \) |
| 97 | \( 1 + 0.793iT - 97T^{2} \) |
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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
−11.95873023234999999496642164353, −11.11626298913432361959307357979, −10.60253809772373046624706830647, −9.738475270044842808658167959889, −7.43735425662497186247671630949, −6.61186440811528693520637865728, −5.96281867509425510528515656689, −4.93289188359974662088456373563, −3.84281614992985518959048462010, −1.90726751443900646424318665186,
1.04206969653358669503518575696, 3.74211996645465078194148442757, 4.83473300905408048658559794427, 5.69407752979911591615350238167, 6.25756546181465073780046686618, 7.49575372754368627204078456596, 8.759413878605225578253350509719, 10.41483633511332557054976182408, 11.14061928289607826218903444498, 11.81122338368268227551247342240