L(s) = 1 | + (1.28 + 1.28i)2-s + (−2.65 + 1.53i)3-s + 1.30i·4-s + (1.07 + 0.288i)5-s + (−5.38 − 1.44i)6-s + (−0.978 + 2.45i)7-s + (0.899 − 0.899i)8-s + (3.21 − 5.56i)9-s + (1.01 + 1.75i)10-s + (0.124 + 0.0332i)11-s + (−1.99 − 3.45i)12-s + (3.59 − 0.291i)13-s + (−4.41 + 1.90i)14-s + (−3.30 + 0.884i)15-s + 4.90·16-s − 0.273·17-s + ⋯ |
L(s) = 1 | + (0.908 + 0.908i)2-s + (−1.53 + 0.886i)3-s + 0.650i·4-s + (0.480 + 0.128i)5-s + (−2.19 − 0.589i)6-s + (−0.369 + 0.929i)7-s + (0.317 − 0.317i)8-s + (1.07 − 1.85i)9-s + (0.319 + 0.553i)10-s + (0.0374 + 0.0100i)11-s + (−0.576 − 0.997i)12-s + (0.996 − 0.0807i)13-s + (−1.17 + 0.508i)14-s + (−0.852 + 0.228i)15-s + 1.22·16-s − 0.0663·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.369 - 0.929i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.369 - 0.929i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.598216 + 0.881438i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.598216 + 0.881438i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 + (0.978 - 2.45i)T \) |
| 13 | \( 1 + (-3.59 + 0.291i)T \) |
good | 2 | \( 1 + (-1.28 - 1.28i)T + 2iT^{2} \) |
| 3 | \( 1 + (2.65 - 1.53i)T + (1.5 - 2.59i)T^{2} \) |
| 5 | \( 1 + (-1.07 - 0.288i)T + (4.33 + 2.5i)T^{2} \) |
| 11 | \( 1 + (-0.124 - 0.0332i)T + (9.52 + 5.5i)T^{2} \) |
| 17 | \( 1 + 0.273T + 17T^{2} \) |
| 19 | \( 1 + (1.02 + 3.83i)T + (-16.4 + 9.5i)T^{2} \) |
| 23 | \( 1 + 0.491iT - 23T^{2} \) |
| 29 | \( 1 + (4.62 - 8.00i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (1.85 + 6.92i)T + (-26.8 + 15.5i)T^{2} \) |
| 37 | \( 1 + (2.82 - 2.82i)T - 37iT^{2} \) |
| 41 | \( 1 + (2.31 + 8.63i)T + (-35.5 + 20.5i)T^{2} \) |
| 43 | \( 1 + (-8.44 + 4.87i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (1.51 - 5.66i)T + (-40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (-0.467 + 0.809i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-4.86 - 4.86i)T + 59iT^{2} \) |
| 61 | \( 1 + (-6.74 - 3.89i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (0.316 - 1.18i)T + (-58.0 - 33.5i)T^{2} \) |
| 71 | \( 1 + (0.793 - 2.96i)T + (-61.4 - 35.5i)T^{2} \) |
| 73 | \( 1 + (-0.118 + 0.0318i)T + (63.2 - 36.5i)T^{2} \) |
| 79 | \( 1 + (5.72 + 9.92i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (8.07 - 8.07i)T - 83iT^{2} \) |
| 89 | \( 1 + (-2.99 - 2.99i)T + 89iT^{2} \) |
| 97 | \( 1 + (-2.39 - 0.641i)T + (84.0 + 48.5i)T^{2} \) |
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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
−14.74851450397364429271436943976, −13.37052385633610508226339289241, −12.43016801701828901024262318633, −11.25530951064230577806938249700, −10.27727499101005155564501818332, −9.089175725530556560409572733438, −6.84866600797019625432923372201, −5.88869227315616747360650502528, −5.37694337279331595564383384257, −3.99068983527187667192597091184,
1.55242273357904087163689461817, 3.96931734486736805216553624280, 5.44227355148486047448044757220, 6.40512464420845354367786730515, 7.76692365001041201194458053001, 10.08346489801623670842740643305, 10.99890995513393708429242989462, 11.68619548305655582608573915381, 12.81772120653903552528501096838, 13.24528483423798768008905810802