| L(s) = 1 | + (0.0723 + 1.99i)2-s + (−1.67 + 0.448i)3-s + (−3.98 + 0.289i)4-s + (4.92 + 1.31i)5-s + (−1.01 − 3.31i)6-s + (4.79 + 5.09i)7-s + (−0.866 − 7.95i)8-s + (2.59 − 1.50i)9-s + (−2.27 + 9.92i)10-s + (21.1 − 5.66i)11-s + (6.54 − 2.27i)12-s + (−3.69 − 3.69i)13-s + (−9.84 + 9.95i)14-s − 8.82·15-s + (15.8 − 2.30i)16-s + (10.5 + 6.08i)17-s + ⋯ |
| L(s) = 1 | + (0.0361 + 0.999i)2-s + (−0.557 + 0.149i)3-s + (−0.997 + 0.0723i)4-s + (0.984 + 0.263i)5-s + (−0.169 − 0.551i)6-s + (0.685 + 0.728i)7-s + (−0.108 − 0.994i)8-s + (0.288 − 0.166i)9-s + (−0.227 + 0.992i)10-s + (1.92 − 0.514i)11-s + (0.545 − 0.189i)12-s + (−0.284 − 0.284i)13-s + (−0.702 + 0.711i)14-s − 0.588·15-s + (0.989 − 0.144i)16-s + (0.620 + 0.357i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.363 - 0.931i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.363 - 0.931i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.975350 + 1.42760i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.975350 + 1.42760i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.0723 - 1.99i)T \) |
| 3 | \( 1 + (1.67 - 0.448i)T \) |
| 7 | \( 1 + (-4.79 - 5.09i)T \) |
| good | 5 | \( 1 + (-4.92 - 1.31i)T + (21.6 + 12.5i)T^{2} \) |
| 11 | \( 1 + (-21.1 + 5.66i)T + (104. - 60.5i)T^{2} \) |
| 13 | \( 1 + (3.69 + 3.69i)T + 169iT^{2} \) |
| 17 | \( 1 + (-10.5 - 6.08i)T + (144.5 + 250. i)T^{2} \) |
| 19 | \( 1 + (2.59 - 9.66i)T + (-312. - 180.5i)T^{2} \) |
| 23 | \( 1 + (10.4 - 6.04i)T + (264.5 - 458. i)T^{2} \) |
| 29 | \( 1 + (21.8 - 21.8i)T - 841iT^{2} \) |
| 31 | \( 1 + (-28.4 - 16.4i)T + (480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + (-8.40 + 31.3i)T + (-1.18e3 - 684.5i)T^{2} \) |
| 41 | \( 1 + 58.6T + 1.68e3T^{2} \) |
| 43 | \( 1 + (-34.1 - 34.1i)T + 1.84e3iT^{2} \) |
| 47 | \( 1 + (-16.5 + 9.56i)T + (1.10e3 - 1.91e3i)T^{2} \) |
| 53 | \( 1 + (43.3 - 11.6i)T + (2.43e3 - 1.40e3i)T^{2} \) |
| 59 | \( 1 + (-8.13 - 30.3i)T + (-3.01e3 + 1.74e3i)T^{2} \) |
| 61 | \( 1 + (-10.3 + 38.5i)T + (-3.22e3 - 1.86e3i)T^{2} \) |
| 67 | \( 1 + (-0.332 - 1.24i)T + (-3.88e3 + 2.24e3i)T^{2} \) |
| 71 | \( 1 + 88.3iT - 5.04e3T^{2} \) |
| 73 | \( 1 + (-44.3 + 76.8i)T + (-2.66e3 - 4.61e3i)T^{2} \) |
| 79 | \( 1 + (67.8 + 117. i)T + (-3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 + (-81.6 - 81.6i)T + 6.88e3iT^{2} \) |
| 89 | \( 1 + (20.7 + 36.0i)T + (-3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 - 37.3iT - 9.40e3T^{2} \) |
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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
−11.79246930441726276719347834729, −10.52970091466874130514883710322, −9.535693910546920733319793128551, −8.886045471171459340411248094474, −7.75323843114044076855098948495, −6.39721001309733637368030125367, −5.97867967688113029380529442518, −5.01362824254923668770066328090, −3.68538180757053546807161376155, −1.47970535500058024671808660320,
1.05298119196317841800600316665, 2.00279392199509332223956712411, 3.97046773621196683108153765040, 4.79962608926618280397699293645, 5.97242991222181977465450948192, 7.12478004499974835637125530697, 8.511665783328285547116734481626, 9.678236023364418058261093850157, 9.961700579504203277456396293255, 11.31621615141982304579608166338
