L(s) = 1 | + 1.73·7-s + 1.73i·13-s − i·19-s − 25-s + 1.73i·37-s − 2i·43-s + 1.99·49-s − 1.73i·61-s + i·67-s − 73-s + 1.73·79-s + 2.99i·91-s + 97-s − 1.73·103-s + ⋯ |
L(s) = 1 | + 1.73·7-s + 1.73i·13-s − i·19-s − 25-s + 1.73i·37-s − 2i·43-s + 1.99·49-s − 1.73i·61-s + i·67-s − 73-s + 1.73·79-s + 2.99i·91-s + 97-s − 1.73·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.965 - 0.258i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.965 - 0.258i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.346821377\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.346821377\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + T^{2} \) |
| 7 | \( 1 - 1.73T + T^{2} \) |
| 11 | \( 1 + T^{2} \) |
| 13 | \( 1 - 1.73iT - T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 + iT - T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 + T^{2} \) |
| 31 | \( 1 + T^{2} \) |
| 37 | \( 1 - 1.73iT - T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 + 2iT - T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 + T^{2} \) |
| 59 | \( 1 + T^{2} \) |
| 61 | \( 1 + 1.73iT - T^{2} \) |
| 67 | \( 1 - iT - T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + T + T^{2} \) |
| 79 | \( 1 - 1.73T + T^{2} \) |
| 83 | \( 1 + T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 - T + T^{2} \) |
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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
−9.392192049554450542858585355197, −8.726559366612716346686431248580, −8.032262626150231921068616874432, −7.19853684935381494017709774935, −6.45943846897056164066551919052, −5.26162929618539286342752890034, −4.65168210636875656123279438289, −3.87590463956139070015727376229, −2.31466449259314218333480453223, −1.53209581930402943391581363298,
1.27374928404865229311054650625, 2.38737495484767966059098893467, 3.63427805708805487319670145154, 4.59546122557385217629789820771, 5.45725681129531527842857408575, 6.00036035500522940460046212826, 7.49991922128041286452351276793, 7.86786542220392293581463568578, 8.450131488255101074234254138421, 9.485588673504898886092293388365