L(s) = 1 | + i·2-s − 3-s − 4-s + i·5-s − i·6-s + 1.35i·7-s − i·8-s + 9-s − 10-s + 3.19i·11-s + 12-s − 1.35·14-s − i·15-s + 16-s + 2.04·17-s + i·18-s + ⋯ |
L(s) = 1 | + 0.707i·2-s − 0.577·3-s − 0.5·4-s + 0.447i·5-s − 0.408i·6-s + 0.512i·7-s − 0.353i·8-s + 0.333·9-s − 0.316·10-s + 0.964i·11-s + 0.288·12-s − 0.362·14-s − 0.258i·15-s + 0.250·16-s + 0.496·17-s + 0.235i·18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.691 - 0.722i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5070 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.691 - 0.722i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.603976804\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.603976804\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - iT \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 - iT \) |
| 13 | \( 1 \) |
good | 7 | \( 1 - 1.35iT - 7T^{2} \) |
| 11 | \( 1 - 3.19iT - 11T^{2} \) |
| 17 | \( 1 - 2.04T + 17T^{2} \) |
| 19 | \( 1 - 5.34iT - 19T^{2} \) |
| 23 | \( 1 - 8.32T + 23T^{2} \) |
| 29 | \( 1 - 8.07T + 29T^{2} \) |
| 31 | \( 1 - 0.185iT - 31T^{2} \) |
| 37 | \( 1 + 2.46iT - 37T^{2} \) |
| 41 | \( 1 + 12.2iT - 41T^{2} \) |
| 43 | \( 1 - 2.91T + 43T^{2} \) |
| 47 | \( 1 - 13.0iT - 47T^{2} \) |
| 53 | \( 1 - 5.74T + 53T^{2} \) |
| 59 | \( 1 + 1.62iT - 59T^{2} \) |
| 61 | \( 1 - 5.28T + 61T^{2} \) |
| 67 | \( 1 + 1.55iT - 67T^{2} \) |
| 71 | \( 1 - 2.32iT - 71T^{2} \) |
| 73 | \( 1 + 5.11iT - 73T^{2} \) |
| 79 | \( 1 + 1.42T + 79T^{2} \) |
| 83 | \( 1 + 11.1iT - 83T^{2} \) |
| 89 | \( 1 - 3.37iT - 89T^{2} \) |
| 97 | \( 1 - 4.74iT - 97T^{2} \) |
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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
−8.433546993558595963182479100832, −7.47145852076375266823940343542, −7.15004563762755285066587859670, −6.29556373872398008481587689932, −5.69912907947969463471966322563, −5.00586386337691403088419560195, −4.26540727912016278546452516353, −3.29711475573471965272622547439, −2.24504212541803203646051151749, −1.00434870795215488568264849062,
0.67699168484727859841905461558, 1.12064687734971819372143746988, 2.61244253344135603191622391879, 3.32869962824674179506218151530, 4.31049750589098281216614589243, 4.96429630381651544201914169697, 5.55684126284170420093762051828, 6.56874886863863816810710716006, 7.13467314997478405515940390645, 8.176857685145308346210784499726