L(s) = 1 | + (0.707 + 0.707i)2-s + (0.292 − 1.70i)3-s + 1.00i·4-s + (−1.52 + 1.63i)5-s + (1.41 − 0.999i)6-s + (−1.33 + 1.33i)7-s + (−0.707 + 0.707i)8-s + (−2.82 − i)9-s + (−2.23 + 0.0743i)10-s − 1.30i·11-s + (1.70 + 0.292i)12-s + (−1.74 − 1.74i)13-s − 1.89·14-s + (2.33 + 3.08i)15-s − 1.00·16-s + (−5.36 − 5.36i)17-s + ⋯ |
L(s) = 1 | + (0.499 + 0.499i)2-s + (0.169 − 0.985i)3-s + 0.500i·4-s + (−0.683 + 0.730i)5-s + (0.577 − 0.408i)6-s + (−0.506 + 0.506i)7-s + (−0.250 + 0.250i)8-s + (−0.942 − 0.333i)9-s + (−0.706 + 0.0234i)10-s − 0.394i·11-s + (0.492 + 0.0845i)12-s + (−0.484 − 0.484i)13-s − 0.506·14-s + (0.604 + 0.796i)15-s − 0.250·16-s + (−1.30 − 1.30i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.636 + 0.771i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.636 + 0.771i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.237615 - 0.504374i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.237615 - 0.504374i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.707 - 0.707i)T \) |
| 3 | \( 1 + (-0.292 + 1.70i)T \) |
| 5 | \( 1 + (1.52 - 1.63i)T \) |
| 31 | \( 1 - T \) |
good | 7 | \( 1 + (1.33 - 1.33i)T - 7iT^{2} \) |
| 11 | \( 1 + 1.30iT - 11T^{2} \) |
| 13 | \( 1 + (1.74 + 1.74i)T + 13iT^{2} \) |
| 17 | \( 1 + (5.36 + 5.36i)T + 17iT^{2} \) |
| 19 | \( 1 + 4.34iT - 19T^{2} \) |
| 23 | \( 1 + (-4.75 + 4.75i)T - 23iT^{2} \) |
| 29 | \( 1 + 5.50T + 29T^{2} \) |
| 37 | \( 1 + (2 - 2i)T - 37iT^{2} \) |
| 41 | \( 1 - 2.81iT - 41T^{2} \) |
| 43 | \( 1 + (-3.50 - 3.50i)T + 43iT^{2} \) |
| 47 | \( 1 + (3.66 + 3.66i)T + 47iT^{2} \) |
| 53 | \( 1 + (6.45 - 6.45i)T - 53iT^{2} \) |
| 59 | \( 1 + 11.3T + 59T^{2} \) |
| 61 | \( 1 - 12.5T + 61T^{2} \) |
| 67 | \( 1 + (-5.52 + 5.52i)T - 67iT^{2} \) |
| 71 | \( 1 - 10.3iT - 71T^{2} \) |
| 73 | \( 1 + (1.87 + 1.87i)T + 73iT^{2} \) |
| 79 | \( 1 - 4.64iT - 79T^{2} \) |
| 83 | \( 1 + (11.2 - 11.2i)T - 83iT^{2} \) |
| 89 | \( 1 + 7.16T + 89T^{2} \) |
| 97 | \( 1 + (1.49 - 1.49i)T - 97iT^{2} \) |
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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
−9.484126722493444285839632008485, −8.709985468236355654470533700901, −7.87514741598117210839861428409, −6.87287124928949503189336582673, −6.75356783564366125746309891157, −5.56785709453318381014801019369, −4.48719833979254558382253204430, −2.95524078843954083905238207636, −2.67463820966066200994088690898, −0.20285975959512310302288593211,
1.88109378179421902557745172717, 3.47679101688059659460507175894, 4.01715961208425156770059332855, 4.78919509855809733609756954018, 5.71951529806530019443391351267, 6.92824861433768183681636641780, 8.011555765140519363937083389893, 8.992676254281357087286142591665, 9.552932359757116542875143935598, 10.45010363772199555053623235492