| L(s) = 1 | + (−0.0142 − 0.107i)3-s + (2.37 + 1.04i)5-s + (−0.709 + 1.52i)7-s + (2.88 − 0.782i)9-s + (−3.35 + 4.30i)11-s + (−2.66 + 5.19i)13-s + (0.0784 − 0.268i)15-s + (−0.517 − 0.0392i)17-s + (1.49 − 2.24i)19-s + (0.173 + 0.0542i)21-s + (−1.78 − 0.708i)23-s + (1.15 + 1.27i)25-s + (−0.250 − 0.596i)27-s + (9.05 − 3.60i)29-s + (−3.29 + 0.761i)31-s + ⋯ |
| L(s) = 1 | + (−0.00823 − 0.0618i)3-s + (1.06 + 0.468i)5-s + (−0.268 + 0.576i)7-s + (0.961 − 0.260i)9-s + (−1.01 + 1.29i)11-s + (−0.737 + 1.44i)13-s + (0.0202 − 0.0694i)15-s + (−0.125 − 0.00952i)17-s + (0.341 − 0.514i)19-s + (0.0378 + 0.0118i)21-s + (−0.371 − 0.147i)23-s + (0.231 + 0.254i)25-s + (−0.0481 − 0.114i)27-s + (1.68 − 0.668i)29-s + (−0.591 + 0.136i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 668 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.345 - 0.938i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 668 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.345 - 0.938i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.33500 + 0.931581i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.33500 + 0.931581i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 167 | \( 1 + (-7.49 - 10.5i)T \) |
| good | 3 | \( 1 + (0.0142 + 0.107i)T + (-2.89 + 0.785i)T^{2} \) |
| 5 | \( 1 + (-2.37 - 1.04i)T + (3.36 + 3.69i)T^{2} \) |
| 7 | \( 1 + (0.709 - 1.52i)T + (-4.51 - 5.35i)T^{2} \) |
| 11 | \( 1 + (3.35 - 4.30i)T + (-2.67 - 10.6i)T^{2} \) |
| 13 | \( 1 + (2.66 - 5.19i)T + (-7.60 - 10.5i)T^{2} \) |
| 17 | \( 1 + (0.517 + 0.0392i)T + (16.8 + 2.56i)T^{2} \) |
| 19 | \( 1 + (-1.49 + 2.24i)T + (-7.35 - 17.5i)T^{2} \) |
| 23 | \( 1 + (1.78 + 0.708i)T + (16.7 + 15.7i)T^{2} \) |
| 29 | \( 1 + (-9.05 + 3.60i)T + (21.0 - 19.9i)T^{2} \) |
| 31 | \( 1 + (3.29 - 0.761i)T + (27.8 - 13.6i)T^{2} \) |
| 37 | \( 1 + (-6.35 - 1.72i)T + (31.9 + 18.7i)T^{2} \) |
| 41 | \( 1 + (10.1 - 4.96i)T + (25.2 - 32.3i)T^{2} \) |
| 43 | \( 1 + (-0.0217 + 0.229i)T + (-42.2 - 8.08i)T^{2} \) |
| 47 | \( 1 + (-0.0934 - 4.93i)T + (-46.9 + 1.77i)T^{2} \) |
| 53 | \( 1 + (0.741 + 4.31i)T + (-49.9 + 17.7i)T^{2} \) |
| 59 | \( 1 + (-11.2 + 0.851i)T + (58.3 - 8.89i)T^{2} \) |
| 61 | \( 1 + (-4.47 + 4.55i)T + (-1.15 - 60.9i)T^{2} \) |
| 67 | \( 1 + (-2.01 + 0.889i)T + (45.0 - 49.5i)T^{2} \) |
| 71 | \( 1 + (-12.5 + 1.43i)T + (69.1 - 15.9i)T^{2} \) |
| 73 | \( 1 + (-5.08 + 3.81i)T + (20.4 - 70.0i)T^{2} \) |
| 79 | \( 1 + (-1.89 + 5.03i)T + (-59.4 - 52.0i)T^{2} \) |
| 83 | \( 1 + (2.31 - 3.21i)T + (-26.2 - 78.7i)T^{2} \) |
| 89 | \( 1 + (1.63 + 5.59i)T + (-75.0 + 47.8i)T^{2} \) |
| 97 | \( 1 + (3.78 + 0.874i)T + (87.1 + 42.5i)T^{2} \) |
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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
−10.32662477338449629080103553877, −9.784698258081373760863736628337, −9.368719594290440257180586022882, −7.981233782728475082493242256903, −6.86521054625125431653299030235, −6.50878009040419627171841587570, −5.15027918530118212067717915098, −4.38024901499197346016943761661, −2.60746806574584819654998400353, −1.92726967357748401061846089340,
0.887364560677356927653949719305, 2.46669158983916479297365436848, 3.67355443589795045149733476578, 5.17023134977348245719852424222, 5.55508483815948605648759861615, 6.79582722897956547857981593112, 7.81112722692765877141396495694, 8.540795243951943286295982456566, 9.841501298855339620196367198366, 10.18462693157782036333938349131
