L(s) = 1 | + i·3-s − 9-s + 4·11-s − 2i·13-s + 2i·17-s + 4·19-s − 8i·23-s − i·27-s + 6·29-s − 8·31-s + 4i·33-s − 6i·37-s + 2·39-s − 6·41-s − 4i·43-s + ⋯ |
L(s) = 1 | + 0.577i·3-s − 0.333·9-s + 1.20·11-s − 0.554i·13-s + 0.485i·17-s + 0.917·19-s − 1.66i·23-s − 0.192i·27-s + 1.11·29-s − 1.43·31-s + 0.696i·33-s − 0.986i·37-s + 0.320·39-s − 0.937·41-s − 0.609i·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.928846232\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.928846232\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 \) |
good | 7 | \( 1 - 7T^{2} \) |
| 11 | \( 1 - 4T + 11T^{2} \) |
| 13 | \( 1 + 2iT - 13T^{2} \) |
| 17 | \( 1 - 2iT - 17T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 23 | \( 1 + 8iT - 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 + 8T + 31T^{2} \) |
| 37 | \( 1 + 6iT - 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 + 4iT - 43T^{2} \) |
| 47 | \( 1 - 47T^{2} \) |
| 53 | \( 1 + 2iT - 53T^{2} \) |
| 59 | \( 1 + 4T + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 + 4iT - 67T^{2} \) |
| 71 | \( 1 + 8T + 71T^{2} \) |
| 73 | \( 1 + 10iT - 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 - 4iT - 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 - 2iT - 97T^{2} \) |
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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
−8.424117643627224854072035486682, −7.49034964923316224394793039772, −6.73992421422624220313852311772, −6.01942508240486101535535863172, −5.27339774414925197941804127911, −4.43299961239515120660455410705, −3.74177384615162290237294733390, −2.98009446876408973938022684240, −1.84474870040763547816016258726, −0.59958430187224091761113230571,
1.08681479381187811855056699751, 1.76586267856129491006685400184, 2.99831173469169723440320977672, 3.70228623064773503993734888656, 4.65873853352764153486943220445, 5.51748132225908171133744904054, 6.22316178507312343148969544769, 7.07255251982457513104914842911, 7.36096123900720239166722200270, 8.342349172206745293383184814277