L(s) = 1 | + (0.707 − 0.707i)2-s + (0.448 + 1.67i)3-s − 1.00i·4-s + (1.5 + 0.866i)6-s + (2.44 + 2.44i)7-s + (−0.707 − 0.707i)8-s + (−2.59 + 1.50i)9-s − 5.19i·11-s + (1.67 − 0.448i)12-s + 3.46·14-s − 1.00·16-s + (−2.12 + 2.12i)17-s + (−0.776 + 2.89i)18-s − i·19-s + (−3 + 5.19i)21-s + (−3.67 − 3.67i)22-s + ⋯ |
L(s) = 1 | + (0.499 − 0.499i)2-s + (0.258 + 0.965i)3-s − 0.500i·4-s + (0.612 + 0.353i)6-s + (0.925 + 0.925i)7-s + (−0.250 − 0.250i)8-s + (−0.866 + 0.5i)9-s − 1.56i·11-s + (0.482 − 0.129i)12-s + 0.925·14-s − 0.250·16-s + (−0.514 + 0.514i)17-s + (−0.183 + 0.683i)18-s − 0.229i·19-s + (−0.654 + 1.13i)21-s + (−0.783 − 0.783i)22-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 - 0.130i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.991 - 0.130i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.53972 + 0.100729i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.53972 + 0.100729i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-0.448 - 1.67i)T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + (-2.44 - 2.44i)T + 7iT^{2} \) |
| 11 | \( 1 + 5.19iT - 11T^{2} \) |
| 13 | \( 1 - 13iT^{2} \) |
| 17 | \( 1 + (2.12 - 2.12i)T - 17iT^{2} \) |
| 19 | \( 1 + iT - 19T^{2} \) |
| 23 | \( 1 + (4.24 + 4.24i)T + 23iT^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 + 2T + 31T^{2} \) |
| 37 | \( 1 + (2.44 + 2.44i)T + 37iT^{2} \) |
| 41 | \( 1 - 5.19iT - 41T^{2} \) |
| 43 | \( 1 + (2.44 - 2.44i)T - 43iT^{2} \) |
| 47 | \( 1 - 47iT^{2} \) |
| 53 | \( 1 + (-4.24 - 4.24i)T + 53iT^{2} \) |
| 59 | \( 1 - 10.3T + 59T^{2} \) |
| 61 | \( 1 - 14T + 61T^{2} \) |
| 67 | \( 1 + (-3.67 - 3.67i)T + 67iT^{2} \) |
| 71 | \( 1 - 71T^{2} \) |
| 73 | \( 1 + (6.12 - 6.12i)T - 73iT^{2} \) |
| 79 | \( 1 + 14iT - 79T^{2} \) |
| 83 | \( 1 + (-2.12 - 2.12i)T + 83iT^{2} \) |
| 89 | \( 1 + 15.5T + 89T^{2} \) |
| 97 | \( 1 + (-4.89 - 4.89i)T + 97iT^{2} \) |
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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
−13.11640569498098506364431003441, −11.71764055761071278198965490687, −11.16978184615749421605430013517, −10.21795996725895416036026937934, −8.840530825379909451702300437125, −8.318738013193146575953123488860, −6.05292294595787903370532722539, −5.14095023665045280523795529715, −3.87838820991488060311248225288, −2.47033492955680976088709044635,
1.96713013012517432355421949032, 4.00531687195082858702010787900, 5.29478349274904067171114416408, 6.90561952511872856444267621648, 7.42801791092927869062854654229, 8.425235377073932184631894571370, 9.882036399215379994954478530424, 11.34725784440517683477289839929, 12.19016416029915654248787947267, 13.17192093826113317992082671663