L(s) = 1 | + 1.73i·3-s + (1.41 − 1.73i)5-s − 11-s + 1.73i·13-s + (2.99 + 2.44i)15-s + 5.19i·17-s + 2.82·19-s − 2.44i·23-s + (−0.999 − 4.89i)25-s + 5.19i·27-s + 7·29-s + 7.07·31-s − 1.73i·33-s − 7.34i·37-s − 2.99·39-s + ⋯ |
L(s) = 1 | + 0.999i·3-s + (0.632 − 0.774i)5-s − 0.301·11-s + 0.480i·13-s + (0.774 + 0.632i)15-s + 1.26i·17-s + 0.648·19-s − 0.510i·23-s + (−0.199 − 0.979i)25-s + 1.00i·27-s + 1.29·29-s + 1.27·31-s − 0.301i·33-s − 1.20i·37-s − 0.480·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.632 - 0.774i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.632 - 0.774i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.67839 + 0.796395i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.67839 + 0.796395i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (-1.41 + 1.73i)T \) |
| 7 | \( 1 \) |
good | 3 | \( 1 - 1.73iT - 3T^{2} \) |
| 11 | \( 1 + T + 11T^{2} \) |
| 13 | \( 1 - 1.73iT - 13T^{2} \) |
| 17 | \( 1 - 5.19iT - 17T^{2} \) |
| 19 | \( 1 - 2.82T + 19T^{2} \) |
| 23 | \( 1 + 2.44iT - 23T^{2} \) |
| 29 | \( 1 - 7T + 29T^{2} \) |
| 31 | \( 1 - 7.07T + 31T^{2} \) |
| 37 | \( 1 + 7.34iT - 37T^{2} \) |
| 41 | \( 1 - 7.07T + 41T^{2} \) |
| 43 | \( 1 - 9.79iT - 43T^{2} \) |
| 47 | \( 1 - 12.1iT - 47T^{2} \) |
| 53 | \( 1 + 12.2iT - 53T^{2} \) |
| 59 | \( 1 - 7.07T + 59T^{2} \) |
| 61 | \( 1 + 14.1T + 61T^{2} \) |
| 67 | \( 1 - 12.2iT - 67T^{2} \) |
| 71 | \( 1 + 10T + 71T^{2} \) |
| 73 | \( 1 - 73T^{2} \) |
| 79 | \( 1 + 3T + 79T^{2} \) |
| 83 | \( 1 - 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 + 5.19iT - 97T^{2} \) |
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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
−10.06584435238686287131913165449, −9.403520240239889306758501129886, −8.659121205937978310879968849422, −7.82551501561621707702027970859, −6.49841297282546506369378869593, −5.70311571367136857598680613055, −4.65492401667521883578483007137, −4.20130493206798685793526296188, −2.78712103641037275631822818127, −1.33550535515666630077581757046,
1.02963108604203505102248180128, 2.37650688826741982875389195836, 3.14892450851111631761051724016, 4.75634776953072748634218040842, 5.77165088977816873192129970274, 6.62158205562783857525839254366, 7.29023677473460190581143612637, 7.934421282122085477851773592364, 9.087127889870169713504745188712, 10.03355762658690381931632145036