| L(s) = 1 | + 1.93i·2-s − 1.73·3-s + 0.273·4-s + (4.88 − 1.06i)5-s − 3.34i·6-s + (−0.433 + 6.98i)7-s + 8.24i·8-s + 2.99·9-s + (2.04 + 9.43i)10-s + 4.41·11-s − 0.473·12-s − 17.0·13-s + (−13.4 − 0.836i)14-s + (−8.46 + 1.83i)15-s − 14.8·16-s + 18.6·17-s + ⋯ |
| L(s) = 1 | + 0.965i·2-s − 0.577·3-s + 0.0683·4-s + (0.977 − 0.212i)5-s − 0.557i·6-s + (−0.0619 + 0.998i)7-s + 1.03i·8-s + 0.333·9-s + (0.204 + 0.943i)10-s + 0.401·11-s − 0.0394·12-s − 1.31·13-s + (−0.963 − 0.0597i)14-s + (−0.564 + 0.122i)15-s − 0.927·16-s + 1.09·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.151 - 0.988i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.151 - 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.919407 + 1.07077i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.919407 + 1.07077i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 1.73T \) |
| 5 | \( 1 + (-4.88 + 1.06i)T \) |
| 7 | \( 1 + (0.433 - 6.98i)T \) |
| good | 2 | \( 1 - 1.93iT - 4T^{2} \) |
| 11 | \( 1 - 4.41T + 121T^{2} \) |
| 13 | \( 1 + 17.0T + 169T^{2} \) |
| 17 | \( 1 - 18.6T + 289T^{2} \) |
| 19 | \( 1 - 18.4iT - 361T^{2} \) |
| 23 | \( 1 + 33.8iT - 529T^{2} \) |
| 29 | \( 1 - 23.8T + 841T^{2} \) |
| 31 | \( 1 + 2.78iT - 961T^{2} \) |
| 37 | \( 1 + 61.7iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 53.1iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 7.46iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 44.3T + 2.20e3T^{2} \) |
| 53 | \( 1 + 30.0iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 19.9iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 64.6iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 68.3iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 14.6T + 5.04e3T^{2} \) |
| 73 | \( 1 - 59.4T + 5.32e3T^{2} \) |
| 79 | \( 1 + 57.2T + 6.24e3T^{2} \) |
| 83 | \( 1 - 98.3T + 6.88e3T^{2} \) |
| 89 | \( 1 - 64.6iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 182.T + 9.40e3T^{2} \) |
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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
−14.34645380536546291602342178419, −12.54161095410854648824149323171, −12.02861634477388988996074112235, −10.50023685123249038780883000466, −9.433727670756919398385303288143, −8.181557974592834353036168949326, −6.79994767138842194458918083519, −5.84651384891135233563033750580, −5.08466545923262821819997442544, −2.27637711253224263927652450640,
1.30876033702465944691694840647, 3.08481543213836264809637131116, 4.85418818238652379313715802368, 6.45658028192918292070863659346, 7.38051298733488770815053275339, 9.712854637041217907919688412666, 10.01205158606021840496607901469, 11.13886569544986865039476877385, 12.01434304550166355740712738243, 13.07282308759185927473410850064
