L(s) = 1 | + (0.257 − 0.446i)3-s + (−1.09 − 1.90i)5-s + (−2.48 + 0.908i)7-s + (1.36 + 2.36i)9-s + (−2.11 + 3.65i)11-s + 3.63·13-s − 1.13·15-s + (3.99 − 6.91i)17-s + (−0.570 − 0.987i)19-s + (−0.234 + 1.34i)21-s + (3.71 + 6.43i)23-s + (0.0808 − 0.139i)25-s + 2.95·27-s + 6.08·29-s + (0.666 − 1.15i)31-s + ⋯ |
L(s) = 1 | + (0.148 − 0.257i)3-s + (−0.491 − 0.851i)5-s + (−0.939 + 0.343i)7-s + (0.455 + 0.789i)9-s + (−0.636 + 1.10i)11-s + 1.00·13-s − 0.292·15-s + (0.968 − 1.67i)17-s + (−0.130 − 0.226i)19-s + (−0.0512 + 0.292i)21-s + (0.775 + 1.34i)23-s + (0.0161 − 0.0279i)25-s + 0.568·27-s + 1.12·29-s + (0.119 − 0.207i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.972 + 0.234i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.972 + 0.234i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.485356701\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.485356701\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 + (2.48 - 0.908i)T \) |
| 41 | \( 1 + T \) |
good | 3 | \( 1 + (-0.257 + 0.446i)T + (-1.5 - 2.59i)T^{2} \) |
| 5 | \( 1 + (1.09 + 1.90i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (2.11 - 3.65i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 - 3.63T + 13T^{2} \) |
| 17 | \( 1 + (-3.99 + 6.91i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (0.570 + 0.987i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-3.71 - 6.43i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 - 6.08T + 29T^{2} \) |
| 31 | \( 1 + (-0.666 + 1.15i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-1.43 - 2.49i)T + (-18.5 + 32.0i)T^{2} \) |
| 43 | \( 1 - 5.14T + 43T^{2} \) |
| 47 | \( 1 + (4.06 + 7.03i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-0.815 + 1.41i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-1.48 + 2.58i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (5.13 + 8.89i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (1.23 - 2.13i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 13.0T + 71T^{2} \) |
| 73 | \( 1 + (5.75 - 9.97i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (4.53 + 7.85i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 7.51T + 83T^{2} \) |
| 89 | \( 1 + (-8.99 - 15.5i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 13.2T + 97T^{2} \) |
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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
−9.700436037356559039475163564573, −8.994842214323592954578763820402, −8.033381853812970896921997449424, −7.41414987378685441075662706232, −6.58055781612520938262880316982, −5.24522863767893090863511661306, −4.79527980665895277181155352929, −3.49956224804698737534428888457, −2.43843510459598837841489634269, −0.959275471512002081484980633082,
0.921715076445077723872158642443, 3.03879430769504364757075755751, 3.42874911009475053696769739680, 4.31779482572327897223869303910, 6.01781267897362126739947414264, 6.31378230704420726892783201023, 7.30343635052221923560599132730, 8.294792352330389456201743106909, 8.909348589072269713035626630369, 10.08725319122811968913651949917