L(s) = 1 | + (1 + i)2-s − 2.44i·3-s + 2i·4-s + (2.44 − 2.44i)6-s + (1 + 2.44i)7-s + (−2 + 2i)8-s − 2.99·9-s + 5i·11-s + 4.89·12-s + 2.44·13-s + (−1.44 + 3.44i)14-s − 4·16-s + 4.89·17-s + (−2.99 − 2.99i)18-s + (5.99 − 2.44i)21-s + (−5 + 5i)22-s + ⋯ |
L(s) = 1 | + (0.707 + 0.707i)2-s − 1.41i·3-s + i·4-s + (0.999 − 0.999i)6-s + (0.377 + 0.925i)7-s + (−0.707 + 0.707i)8-s − 0.999·9-s + 1.50i·11-s + 1.41·12-s + 0.679·13-s + (−0.387 + 0.921i)14-s − 16-s + 1.18·17-s + (−0.707 − 0.707i)18-s + (1.30 − 0.534i)21-s + (−1.06 + 1.06i)22-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.659 - 0.752i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.659 - 0.752i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.06359 + 0.935493i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.06359 + 0.935493i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1 - i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-1 - 2.44i)T \) |
good | 3 | \( 1 + 2.44iT - 3T^{2} \) |
| 11 | \( 1 - 5iT - 11T^{2} \) |
| 13 | \( 1 - 2.44T + 13T^{2} \) |
| 17 | \( 1 - 4.89T + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 - T + 23T^{2} \) |
| 29 | \( 1 - 5T + 29T^{2} \) |
| 31 | \( 1 + 7.34T + 31T^{2} \) |
| 37 | \( 1 + 3iT - 37T^{2} \) |
| 41 | \( 1 + 12.2iT - 41T^{2} \) |
| 43 | \( 1 - 11T + 43T^{2} \) |
| 47 | \( 1 - 4.89iT - 47T^{2} \) |
| 53 | \( 1 - 4iT - 53T^{2} \) |
| 59 | \( 1 + 12.2T + 59T^{2} \) |
| 61 | \( 1 - 12.2iT - 61T^{2} \) |
| 67 | \( 1 + 3T + 67T^{2} \) |
| 71 | \( 1 + 5iT - 71T^{2} \) |
| 73 | \( 1 - 2.44T + 73T^{2} \) |
| 79 | \( 1 - 9iT - 79T^{2} \) |
| 83 | \( 1 + 2.44iT - 83T^{2} \) |
| 89 | \( 1 + 2.44iT - 89T^{2} \) |
| 97 | \( 1 + 7.34T + 97T^{2} \) |
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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
−10.84440248981711410481022102669, −9.348275415192019322234696948335, −8.513966183572242591876891014490, −7.53704817451877191802886562147, −7.22700156342125919018937832154, −6.08292224185065765233098561705, −5.47538390635044159896760607627, −4.26257573306103489844245627465, −2.76551046586523692109272567895, −1.71281013558769763944698737375,
1.08183980486081166775265430359, 3.19435017120298502417566485818, 3.67607109868966331122676514392, 4.64387180280913318019407866782, 5.47577122309862483672669103136, 6.38167657375934576528634393927, 7.907262127951339798432702711968, 8.953592512391985508098437863434, 9.776390391608005098609560001172, 10.58907166253895625077019801107