| L(s) = 1 | + (−0.981 + 0.189i)2-s + (−1.08 + 2.38i)3-s + (0.928 − 0.371i)4-s + (−2.17 − 0.637i)5-s + (0.617 − 2.54i)6-s + (−0.589 + 1.70i)7-s + (−0.841 + 0.540i)8-s + (−2.53 − 2.92i)9-s + (2.25 + 0.215i)10-s + (−1.09 − 4.50i)11-s + (−0.124 + 2.61i)12-s + (−5.15 + 2.65i)13-s + (0.256 − 1.78i)14-s + (3.88 − 4.47i)15-s + (0.723 − 0.690i)16-s + (−1.84 − 0.739i)17-s + ⋯ |
| L(s) = 1 | + (−0.694 + 0.133i)2-s + (−0.628 + 1.37i)3-s + (0.464 − 0.185i)4-s + (−0.970 − 0.285i)5-s + (0.252 − 1.03i)6-s + (−0.222 + 0.643i)7-s + (−0.297 + 0.191i)8-s + (−0.843 − 0.973i)9-s + (0.712 + 0.0680i)10-s + (−0.329 − 1.35i)11-s + (−0.0359 + 0.755i)12-s + (−1.43 + 0.737i)13-s + (0.0685 − 0.476i)14-s + (1.00 − 1.15i)15-s + (0.180 − 0.172i)16-s + (−0.448 − 0.179i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 134 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.995 + 0.0964i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 134 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.995 + 0.0964i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0134279 - 0.277896i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0134279 - 0.277896i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.981 - 0.189i)T \) |
| 67 | \( 1 + (-0.800 + 8.14i)T \) |
| good | 3 | \( 1 + (1.08 - 2.38i)T + (-1.96 - 2.26i)T^{2} \) |
| 5 | \( 1 + (2.17 + 0.637i)T + (4.20 + 2.70i)T^{2} \) |
| 7 | \( 1 + (0.589 - 1.70i)T + (-5.50 - 4.32i)T^{2} \) |
| 11 | \( 1 + (1.09 + 4.50i)T + (-9.77 + 5.04i)T^{2} \) |
| 13 | \( 1 + (5.15 - 2.65i)T + (7.54 - 10.5i)T^{2} \) |
| 17 | \( 1 + (1.84 + 0.739i)T + (12.3 + 11.7i)T^{2} \) |
| 19 | \( 1 + (-2.40 - 6.95i)T + (-14.9 + 11.7i)T^{2} \) |
| 23 | \( 1 + (-3.96 - 5.57i)T + (-7.52 + 21.7i)T^{2} \) |
| 29 | \( 1 + (-0.844 - 1.46i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-3.86 - 1.99i)T + (17.9 + 25.2i)T^{2} \) |
| 37 | \( 1 + (0.411 - 0.712i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (4.15 - 3.27i)T + (9.66 - 39.8i)T^{2} \) |
| 43 | \( 1 + (0.942 + 6.55i)T + (-41.2 + 12.1i)T^{2} \) |
| 47 | \( 1 + (-0.829 + 0.0791i)T + (46.1 - 8.89i)T^{2} \) |
| 53 | \( 1 + (1.06 - 7.41i)T + (-50.8 - 14.9i)T^{2} \) |
| 59 | \( 1 + (11.3 - 7.31i)T + (24.5 - 53.6i)T^{2} \) |
| 61 | \( 1 + (-0.453 + 1.87i)T + (-54.2 - 27.9i)T^{2} \) |
| 71 | \( 1 + (-2.74 + 1.09i)T + (51.3 - 48.9i)T^{2} \) |
| 73 | \( 1 + (-1.87 + 7.73i)T + (-64.8 - 33.4i)T^{2} \) |
| 79 | \( 1 + (0.527 - 11.0i)T + (-78.6 - 7.50i)T^{2} \) |
| 83 | \( 1 + (-3.32 + 3.17i)T + (3.94 - 82.9i)T^{2} \) |
| 89 | \( 1 + (6.74 + 14.7i)T + (-58.2 + 67.2i)T^{2} \) |
| 97 | \( 1 + (-3.70 + 6.42i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−14.05301344948885957850076974576, −12.12226315650475867355893888550, −11.63304570676801563813581615650, −10.61973825227796356860808061996, −9.625884968789730939488076709290, −8.777826090037910891113098848594, −7.58489249462229268012357510936, −5.90511764226826014986793098004, −4.83725047807876230356605228150, −3.36233810179611749656849744935,
0.36723383128971821163913357151, 2.54436820803164887152708230045, 4.79689754257922557657147687969, 6.85131228585263609177474005165, 7.19427347505606919343349488592, 8.031760167782328763433580171628, 9.732230083254256655146360804897, 10.83701178778820909726959216196, 11.78752929426524388781134003258, 12.53629482764477635735039889474
