| L(s) = 1 | + (−0.0581 − 1.21i)2-s + (0.506 − 0.0483i)4-s + (0.949 + 1.09i)5-s + (1.57 − 3.04i)7-s + (−0.435 − 3.03i)8-s + (1.28 − 1.22i)10-s + (−0.207 + 0.599i)11-s + (−0.662 + 0.841i)13-s + (−3.81 − 1.73i)14-s + (−2.67 + 0.515i)16-s + (−0.229 + 2.40i)17-s + (5.75 − 2.96i)19-s + (0.533 + 0.508i)20-s + (0.743 + 0.218i)22-s + (−1.93 − 0.470i)23-s + ⋯ |
| L(s) = 1 | + (−0.0410 − 0.862i)2-s + (0.253 − 0.0241i)4-s + (0.424 + 0.490i)5-s + (0.594 − 1.15i)7-s + (−0.154 − 1.07i)8-s + (0.405 − 0.386i)10-s + (−0.0625 + 0.180i)11-s + (−0.183 + 0.233i)13-s + (−1.01 − 0.465i)14-s + (−0.668 + 0.128i)16-s + (−0.0556 + 0.582i)17-s + (1.32 − 0.681i)19-s + (0.119 + 0.113i)20-s + (0.158 + 0.0465i)22-s + (−0.404 − 0.0980i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 603 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0417 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 603 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0417 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.28385 - 1.33862i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.28385 - 1.33862i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 67 | \( 1 + (4.94 - 6.52i)T \) |
| good | 2 | \( 1 + (0.0581 + 1.21i)T + (-1.99 + 0.190i)T^{2} \) |
| 5 | \( 1 + (-0.949 - 1.09i)T + (-0.711 + 4.94i)T^{2} \) |
| 7 | \( 1 + (-1.57 + 3.04i)T + (-4.06 - 5.70i)T^{2} \) |
| 11 | \( 1 + (0.207 - 0.599i)T + (-8.64 - 6.79i)T^{2} \) |
| 13 | \( 1 + (0.662 - 0.841i)T + (-3.06 - 12.6i)T^{2} \) |
| 17 | \( 1 + (0.229 - 2.40i)T + (-16.6 - 3.21i)T^{2} \) |
| 19 | \( 1 + (-5.75 + 2.96i)T + (11.0 - 15.4i)T^{2} \) |
| 23 | \( 1 + (1.93 + 0.470i)T + (20.4 + 10.5i)T^{2} \) |
| 29 | \( 1 + (5.18 - 2.99i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-3.17 - 4.04i)T + (-7.30 + 30.1i)T^{2} \) |
| 37 | \( 1 + (-0.645 + 1.11i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-4.34 + 6.09i)T + (-13.4 - 38.7i)T^{2} \) |
| 43 | \( 1 + (0.287 - 0.131i)T + (28.1 - 32.4i)T^{2} \) |
| 47 | \( 1 + (-4.56 + 4.78i)T + (-2.23 - 46.9i)T^{2} \) |
| 53 | \( 1 + (1.60 - 3.51i)T + (-34.7 - 40.0i)T^{2} \) |
| 59 | \( 1 + (1.38 - 0.199i)T + (56.6 - 16.6i)T^{2} \) |
| 61 | \( 1 + (12.6 - 4.36i)T + (47.9 - 37.7i)T^{2} \) |
| 71 | \( 1 + (-1.18 - 12.3i)T + (-69.7 + 13.4i)T^{2} \) |
| 73 | \( 1 + (0.711 + 2.05i)T + (-57.3 + 45.1i)T^{2} \) |
| 79 | \( 1 + (-2.83 - 7.09i)T + (-57.1 + 54.5i)T^{2} \) |
| 83 | \( 1 + (1.98 + 10.2i)T + (-77.0 + 30.8i)T^{2} \) |
| 89 | \( 1 + (0.0633 - 0.215i)T + (-74.8 - 48.1i)T^{2} \) |
| 97 | \( 1 + (2.46 + 1.42i)T + (48.5 + 84.0i)T^{2} \) |
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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
−10.49912098493096768337852764199, −9.983309958582079841770819666897, −8.943087962139331019412444121134, −7.50098963480905261376097123173, −7.06150064459289916666522988757, −5.93313552220000785270317384374, −4.55430754403030731379667481319, −3.53406470381158702454289900192, −2.36941671151472973846809248386, −1.16732415290727543837827057922,
1.78514189485287420920284742183, 2.95973514947980278366841882569, 4.82856524834869122915157854478, 5.60986460444033046283151746092, 6.12527345051030285382211998816, 7.55332943766965040431794334510, 7.996031764424509743186432707674, 9.063267117838593265331624327408, 9.668874315608078588803702313882, 11.07870881396970641213110127749
