L(s) = 1 | − 1.41i·2-s − 2.00·4-s + 2.23i·5-s − 1.16·7-s + 2.82i·8-s + 3.16·10-s + 5.88i·11-s − 7.16·13-s + 1.64i·14-s + 4.00·16-s + 6.32·19-s − 4.47i·20-s + 8.32·22-s + 4.47i·23-s − 5.00·25-s + 10.1i·26-s + ⋯ |
L(s) = 1 | − 0.999i·2-s − 1.00·4-s + 0.999i·5-s − 0.439·7-s + 1.00i·8-s + 1.00·10-s + 1.77i·11-s − 1.98·13-s + 0.439i·14-s + 1.00·16-s + 1.45·19-s − 1.00i·20-s + 1.77·22-s + 0.932i·23-s − 1.00·25-s + 1.98i·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 - 0.816i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.711600 + 0.368351i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.711600 + 0.368351i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + 1.41iT \) |
| 3 | \( 1 \) |
| 5 | \( 1 - 2.23iT \) |
good | 7 | \( 1 + 1.16T + 7T^{2} \) |
| 11 | \( 1 - 5.88iT - 11T^{2} \) |
| 13 | \( 1 + 7.16T + 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 - 6.32T + 19T^{2} \) |
| 23 | \( 1 - 4.47iT - 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 - 31T^{2} \) |
| 37 | \( 1 - 4.83T + 37T^{2} \) |
| 41 | \( 1 + 7.53iT - 41T^{2} \) |
| 43 | \( 1 - 43T^{2} \) |
| 47 | \( 1 + 2.82iT - 47T^{2} \) |
| 53 | \( 1 - 5.65iT - 53T^{2} \) |
| 59 | \( 1 - 14.3iT - 59T^{2} \) |
| 61 | \( 1 - 61T^{2} \) |
| 67 | \( 1 - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 73T^{2} \) |
| 79 | \( 1 - 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 - 0.955iT - 89T^{2} \) |
| 97 | \( 1 - 97T^{2} \) |
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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
−11.80405802842352695904409209403, −10.52224586766819960297654973682, −9.726065086764923577699661549181, −9.509454282214263796121856910262, −7.57248417463009744486295346957, −7.17117963633634371830596852850, −5.46393639816286108045392679451, −4.40879456638185582458786883967, −3.07154677028795320041268888923, −2.11346152326012709371305456458,
0.54349211208493593167883578330, 3.19001818160007747045782642096, 4.66845318000949761380246983564, 5.43245148580662300205927331902, 6.42754710439263654696367062392, 7.65054448798978849174210652643, 8.353895079452011656693728009191, 9.380578026218381259992828451154, 9.899874050530552502055687839985, 11.46300341621742172493648226894