L(s) = 1 | − 1.41·3-s − i·5-s + (0.707 + 0.707i)7-s + 1.00·9-s + 1.41i·15-s + (−1.00 − 1.00i)21-s − 1.41i·23-s − 25-s + 2·29-s + (0.707 − 0.707i)35-s − 2i·41-s − 1.41i·43-s − 1.00i·45-s + 1.41·47-s + 1.00i·49-s + ⋯ |
L(s) = 1 | − 1.41·3-s − i·5-s + (0.707 + 0.707i)7-s + 1.00·9-s + 1.41i·15-s + (−1.00 − 1.00i)21-s − 1.41i·23-s − 25-s + 2·29-s + (0.707 − 0.707i)35-s − 2i·41-s − 1.41i·43-s − 1.00i·45-s + 1.41·47-s + 1.00i·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6865920305\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6865920305\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + iT \) |
| 7 | \( 1 + (-0.707 - 0.707i)T \) |
good | 3 | \( 1 + 1.41T + T^{2} \) |
| 11 | \( 1 + T^{2} \) |
| 13 | \( 1 + T^{2} \) |
| 17 | \( 1 + T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 + 1.41iT - T^{2} \) |
| 29 | \( 1 - 2T + T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 + 2iT - T^{2} \) |
| 43 | \( 1 + 1.41iT - T^{2} \) |
| 47 | \( 1 - 1.41T + T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - 2iT - T^{2} \) |
| 67 | \( 1 + 1.41iT - T^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 + T^{2} \) |
| 79 | \( 1 + T^{2} \) |
| 83 | \( 1 + 1.41T + T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 + T^{2} \) |
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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
−10.22436178280597316280247009630, −8.844040108314960212590656204519, −8.571629496588405530900267115376, −7.35451151893149358426022803706, −6.30287041845908139821184006248, −5.57737949411085214059723978377, −4.93017583447507328552005666325, −4.22832342259436788253194371123, −2.33808524479508791222619385669, −0.887202592886606159002293330930,
1.29529814459850154219268865617, 2.95428087845050544134746477136, 4.23054684244487218955244404998, 5.04475919863238095014356382329, 6.01752856155014287296333372906, 6.67544739109877215453841978458, 7.45457185827303044971751324896, 8.263964693136067594504547220583, 9.748849756002115137377545536884, 10.28725779059436993206067943690