L(s) = 1 | + 2i·2-s + 9.74i·3-s − 4·4-s + (−4.75 + 10.1i)5-s − 19.4·6-s − 8i·8-s − 67.9·9-s + (−20.2 − 9.51i)10-s + 6.52·11-s − 38.9i·12-s − 41.6i·13-s + (−98.6 − 46.3i)15-s + 16·16-s + 109. i·17-s − 135. i·18-s + 29.7·19-s + ⋯ |
L(s) = 1 | + 0.707i·2-s + 1.87i·3-s − 0.5·4-s + (−0.425 + 0.905i)5-s − 1.32·6-s − 0.353i·8-s − 2.51·9-s + (−0.639 − 0.300i)10-s + 0.178·11-s − 0.937i·12-s − 0.888i·13-s + (−1.69 − 0.797i)15-s + 0.250·16-s + 1.56i·17-s − 1.78i·18-s + 0.359·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.905 + 0.425i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.905 + 0.425i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.3587897380\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3587897380\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - 2iT \) |
| 5 | \( 1 + (4.75 - 10.1i)T \) |
| 7 | \( 1 \) |
good | 3 | \( 1 - 9.74iT - 27T^{2} \) |
| 11 | \( 1 - 6.52T + 1.33e3T^{2} \) |
| 13 | \( 1 + 41.6iT - 2.19e3T^{2} \) |
| 17 | \( 1 - 109. iT - 4.91e3T^{2} \) |
| 19 | \( 1 - 29.7T + 6.85e3T^{2} \) |
| 23 | \( 1 + 180. iT - 1.21e4T^{2} \) |
| 29 | \( 1 + 183.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 116.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 396. iT - 5.06e4T^{2} \) |
| 41 | \( 1 + 197.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 302. iT - 7.95e4T^{2} \) |
| 47 | \( 1 + 277. iT - 1.03e5T^{2} \) |
| 53 | \( 1 + 405. iT - 1.48e5T^{2} \) |
| 59 | \( 1 + 67.8T + 2.05e5T^{2} \) |
| 61 | \( 1 + 510.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 1.02e3iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 352.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 59.7iT - 3.89e5T^{2} \) |
| 79 | \( 1 - 133.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 571. iT - 5.71e5T^{2} \) |
| 89 | \( 1 - 547.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 783. iT - 9.12e5T^{2} \) |
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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
−10.98388007219051171154100557099, −10.31907363840352090287496092028, −9.833689278338293009395803589877, −8.556988319793063653902178881166, −8.088223333868553287411987240326, −6.57978169193039772203146106252, −5.74949955773848929005328353603, −4.65959755702189590227768481945, −3.81097616264180980857035062942, −2.96381007691675022961302665511,
0.12264614034763938718455466265, 1.19328236535738638054366556766, 2.09741316118424930975066944658, 3.46716939096674654092375483471, 4.97724213882859631443995602340, 5.95588111828386732589757978200, 7.35019520125303305466266586734, 7.63384294308156504564884266401, 8.984604182686364566317715259433, 9.305069117661445509415791672454