L(s) = 1 | + (−0.707 + 0.707i)2-s − 1.00i·4-s + (0.707 + 0.707i)7-s + (0.707 + 0.707i)8-s + 3.41i·11-s + (−2.02 + 2.02i)13-s − 1.00·14-s − 1.00·16-s + (4.37 − 4.37i)17-s − 4.87i·19-s + (−2.41 − 2.41i)22-s + (3.74 + 3.74i)23-s − 2.86i·26-s + (0.707 − 0.707i)28-s + 5.21·29-s + ⋯ |
L(s) = 1 | + (−0.499 + 0.499i)2-s − 0.500i·4-s + (0.267 + 0.267i)7-s + (0.250 + 0.250i)8-s + 1.02i·11-s + (−0.561 + 0.561i)13-s − 0.267·14-s − 0.250·16-s + (1.06 − 1.06i)17-s − 1.11i·19-s + (−0.514 − 0.514i)22-s + (0.780 + 0.780i)23-s − 0.561i·26-s + (0.133 − 0.133i)28-s + 0.968·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.161 - 0.986i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.161 - 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.381882692\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.381882692\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-0.707 - 0.707i)T \) |
good | 11 | \( 1 - 3.41iT - 11T^{2} \) |
| 13 | \( 1 + (2.02 - 2.02i)T - 13iT^{2} \) |
| 17 | \( 1 + (-4.37 + 4.37i)T - 17iT^{2} \) |
| 19 | \( 1 + 4.87iT - 19T^{2} \) |
| 23 | \( 1 + (-3.74 - 3.74i)T + 23iT^{2} \) |
| 29 | \( 1 - 5.21T + 29T^{2} \) |
| 31 | \( 1 + 4.93T + 31T^{2} \) |
| 37 | \( 1 + (-6.87 - 6.87i)T + 37iT^{2} \) |
| 41 | \( 1 + 8.77iT - 41T^{2} \) |
| 43 | \( 1 + (-0.174 + 0.174i)T - 43iT^{2} \) |
| 47 | \( 1 + (3.42 - 3.42i)T - 47iT^{2} \) |
| 53 | \( 1 + (-6.43 - 6.43i)T + 53iT^{2} \) |
| 59 | \( 1 - 2.22T + 59T^{2} \) |
| 61 | \( 1 + 15.2T + 61T^{2} \) |
| 67 | \( 1 + (3.81 + 3.81i)T + 67iT^{2} \) |
| 71 | \( 1 - 10.9iT - 71T^{2} \) |
| 73 | \( 1 + (-2.51 + 2.51i)T - 73iT^{2} \) |
| 79 | \( 1 + 9.98iT - 79T^{2} \) |
| 83 | \( 1 + (-2.35 - 2.35i)T + 83iT^{2} \) |
| 89 | \( 1 - 4.07T + 89T^{2} \) |
| 97 | \( 1 + (-5.64 - 5.64i)T + 97iT^{2} \) |
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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
−9.105888007648190874490852920263, −7.942943811421903208049690477072, −7.30539823187791877485412540897, −6.91442774975638227186657398901, −5.85177446810207224688741861903, −4.96120311613511133279418154484, −4.57194791577040989773107557424, −3.10827393343650203989420938924, −2.18108823679994157617920148507, −0.998958831000997447776626346292,
0.63566087153005768704016647823, 1.65035383270839368215730839981, 2.88911616398679639984627737041, 3.55396518073639348193880556878, 4.49337972918497081270687232821, 5.54876912820462971468465031761, 6.19186115715324552916336473773, 7.24725297545357072437235452397, 8.079649077997763104996973209007, 8.322830110110725385528596257868