L(s) = 1 | + i·2-s − 2.44i·3-s − 4-s + (0.224 + 2.22i)5-s + 2.44·6-s − i·8-s − 2.99·9-s + (−2.22 + 0.224i)10-s + 4.89·11-s + 2.44i·12-s + 0.449i·13-s + (5.44 − 0.550i)15-s + 16-s + 2i·17-s − 2.99i·18-s + 6.44·19-s + ⋯ |
L(s) = 1 | + 0.707i·2-s − 1.41i·3-s − 0.5·4-s + (0.100 + 0.994i)5-s + 0.999·6-s − 0.353i·8-s − 0.999·9-s + (−0.703 + 0.0710i)10-s + 1.47·11-s + 0.707i·12-s + 0.124i·13-s + (1.40 − 0.142i)15-s + 0.250·16-s + 0.485i·17-s − 0.707i·18-s + 1.47·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 - 0.100i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.994 - 0.100i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.49110 + 0.0751248i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.49110 + 0.0751248i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - iT \) |
| 5 | \( 1 + (-0.224 - 2.22i)T \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + 2.44iT - 3T^{2} \) |
| 11 | \( 1 - 4.89T + 11T^{2} \) |
| 13 | \( 1 - 0.449iT - 13T^{2} \) |
| 17 | \( 1 - 2iT - 17T^{2} \) |
| 19 | \( 1 - 6.44T + 19T^{2} \) |
| 23 | \( 1 + 6.89iT - 23T^{2} \) |
| 29 | \( 1 - 2.89T + 29T^{2} \) |
| 31 | \( 1 - 0.898T + 31T^{2} \) |
| 37 | \( 1 + 2iT - 37T^{2} \) |
| 41 | \( 1 - 10.8T + 41T^{2} \) |
| 43 | \( 1 - 8.89iT - 43T^{2} \) |
| 47 | \( 1 + 0.898iT - 47T^{2} \) |
| 53 | \( 1 + 1.10iT - 53T^{2} \) |
| 59 | \( 1 + 6.44T + 59T^{2} \) |
| 61 | \( 1 + 8.44T + 61T^{2} \) |
| 67 | \( 1 - 8iT - 67T^{2} \) |
| 71 | \( 1 + 10.8T + 71T^{2} \) |
| 73 | \( 1 - 6.89iT - 73T^{2} \) |
| 79 | \( 1 - 2.89T + 79T^{2} \) |
| 83 | \( 1 + 2.44iT - 83T^{2} \) |
| 89 | \( 1 + 10T + 89T^{2} \) |
| 97 | \( 1 + 3.79iT - 97T^{2} \) |
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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
−11.17116653368666286274137107340, −9.953602005449436349280177583671, −8.979923032469974288378310796445, −7.906338874764839013728995370057, −7.19236681637688387315842701221, −6.48151124622389409407898717838, −5.95584051141790940636650169868, −4.23042896156416899809120109843, −2.81336266714014496872473979834, −1.26772411955896465249025104503,
1.28296491791730939813381909729, 3.26684341551782606088953771249, 4.12873622388879084565059835855, 4.94241521212648658785106116413, 5.82524168539910398593475117643, 7.53297166757971703566699406231, 8.859846258882581693105894371048, 9.398884655668764404338589796385, 9.799247521051078332184901168992, 10.94606605111687455070055175689