| L(s) = 1 | + (−1.90 − 0.595i)2-s + (3.29 + 2.27i)4-s + 2.67i·5-s + 2.64i·7-s + (−4.92 − 6.30i)8-s + (1.59 − 5.10i)10-s − 8.81·11-s + 0.379i·13-s + (1.57 − 5.05i)14-s + (5.65 + 14.9i)16-s − 21.9·17-s − 5.52·19-s + (−6.07 + 8.79i)20-s + (16.8 + 5.24i)22-s − 19.0i·23-s + ⋯ |
| L(s) = 1 | + (−0.954 − 0.297i)2-s + (0.822 + 0.568i)4-s + 0.534i·5-s + 0.377i·7-s + (−0.616 − 0.787i)8-s + (0.159 − 0.510i)10-s − 0.801·11-s + 0.0291i·13-s + (0.112 − 0.360i)14-s + (0.353 + 0.935i)16-s − 1.29·17-s − 0.290·19-s + (−0.303 + 0.439i)20-s + (0.764 + 0.238i)22-s − 0.828i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.787 + 0.616i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.787 + 0.616i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.2549933334\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2549933334\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.90 + 0.595i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 - 2.64iT \) |
| good | 5 | \( 1 - 2.67iT - 25T^{2} \) |
| 11 | \( 1 + 8.81T + 121T^{2} \) |
| 13 | \( 1 - 0.379iT - 169T^{2} \) |
| 17 | \( 1 + 21.9T + 289T^{2} \) |
| 19 | \( 1 + 5.52T + 361T^{2} \) |
| 23 | \( 1 + 19.0iT - 529T^{2} \) |
| 29 | \( 1 + 15.4iT - 841T^{2} \) |
| 31 | \( 1 + 28.6iT - 961T^{2} \) |
| 37 | \( 1 + 9.53iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 6.73T + 1.68e3T^{2} \) |
| 43 | \( 1 + 3.26T + 1.84e3T^{2} \) |
| 47 | \( 1 + 34.6iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 49.6iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 37.5T + 3.48e3T^{2} \) |
| 61 | \( 1 + 104. iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 119.T + 4.48e3T^{2} \) |
| 71 | \( 1 + 103. iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 112.T + 5.32e3T^{2} \) |
| 79 | \( 1 - 17.4iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 25.6T + 6.88e3T^{2} \) |
| 89 | \( 1 - 16.2T + 7.92e3T^{2} \) |
| 97 | \( 1 + 74.4T + 9.40e3T^{2} \) |
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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
−10.52803629385529794535777097648, −9.438567183823340779217841264839, −8.656120466561074828740834821729, −7.79674968490030917699045576205, −6.83561927661144261388289634871, −6.00936066796790052400950440060, −4.45356010939735991095169800309, −2.95752458793148276953623527854, −2.11417761764942763252867332426, −0.13654387879190115650534720686,
1.41008566425446497091763895950, 2.84296429190912642029071979573, 4.57416012118463158747803166284, 5.58222859612596431729275072842, 6.73258496329356123097868734222, 7.51509616085494362458070290616, 8.531239232877398498778537930426, 9.079006300544947827042096340094, 10.18329223447128480444251994336, 10.80746563374383366072140659917
