L(s) = 1 | + (1.73 − 3.01i)5-s − 3.36i·7-s + 2.43i·11-s + (−5.02 + 2.89i)13-s + (−2.67 + 4.63i)17-s + (−2.84 + 3.30i)19-s + (−1.30 + 0.753i)23-s + (−3.54 − 6.13i)25-s + (4.21 − 2.43i)29-s − 10.5·31-s + (−10.1 − 5.85i)35-s − 8.46i·37-s + (3.41 + 1.96i)41-s + (−9.02 − 5.21i)43-s + (−9.63 + 5.56i)47-s + ⋯ |
L(s) = 1 | + (0.777 − 1.34i)5-s − 1.27i·7-s + 0.733i·11-s + (−1.39 + 0.804i)13-s + (−0.649 + 1.12i)17-s + (−0.652 + 0.757i)19-s + (−0.272 + 0.157i)23-s + (−0.708 − 1.22i)25-s + (0.782 − 0.451i)29-s − 1.89·31-s + (−1.71 − 0.989i)35-s − 1.39i·37-s + (0.532 + 0.307i)41-s + (−1.37 − 0.794i)43-s + (−1.40 + 0.811i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.817 - 0.575i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.817 - 0.575i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.1882422289\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1882422289\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 + (2.84 - 3.30i)T \) |
good | 5 | \( 1 + (-1.73 + 3.01i)T + (-2.5 - 4.33i)T^{2} \) |
| 7 | \( 1 + 3.36iT - 7T^{2} \) |
| 11 | \( 1 - 2.43iT - 11T^{2} \) |
| 13 | \( 1 + (5.02 - 2.89i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (2.67 - 4.63i)T + (-8.5 - 14.7i)T^{2} \) |
| 23 | \( 1 + (1.30 - 0.753i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-4.21 + 2.43i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + 10.5T + 31T^{2} \) |
| 37 | \( 1 + 8.46iT - 37T^{2} \) |
| 41 | \( 1 + (-3.41 - 1.96i)T + (20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (9.02 + 5.21i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (9.63 - 5.56i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-4.41 + 2.54i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (5.58 - 9.66i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (2.80 + 4.86i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-3.03 - 5.26i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-6.15 + 10.6i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (3.97 - 6.88i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-1.19 + 2.06i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 5.59iT - 83T^{2} \) |
| 89 | \( 1 + (-5.04 + 2.91i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-2.47 - 1.43i)T + (48.5 + 84.0i)T^{2} \) |
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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
−8.435359701545463106440021992317, −7.59737073730164394118801779762, −6.91176168958911980037154634657, −6.04285180074669949047451013296, −5.09994370389728753099094123939, −4.40660792616549547214153442335, −3.89949066126554984909456804586, −2.08898882574972479503132030042, −1.58697156926495302727022364876, −0.05343371462235111268870016426,
2.08330482411358380866973197947, 2.70508397674769239452114215880, 3.23894067615354897212020300741, 4.91928937802308153871307379847, 5.37990620169769255764447895834, 6.37972082969656637228942125090, 6.77866418627761279690703214687, 7.70129460086557418932243440312, 8.630510676348692102258616262935, 9.331637468131817696718534724301