L(s) = 1 | + i·2-s − 3-s − 4-s + i·5-s − i·6-s − 2i·7-s − i·8-s + 9-s − 10-s + 0.464i·11-s + 12-s + 2·14-s − i·15-s + 16-s − 4·17-s + i·18-s + ⋯ |
L(s) = 1 | + 0.707i·2-s − 0.577·3-s − 0.5·4-s + 0.447i·5-s − 0.408i·6-s − 0.755i·7-s − 0.353i·8-s + 0.333·9-s − 0.316·10-s + 0.139i·11-s + 0.288·12-s + 0.534·14-s − 0.258i·15-s + 0.250·16-s − 0.970·17-s + 0.235i·18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.277 - 0.960i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5070 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.277 - 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.054056787\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.054056787\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - iT \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 - iT \) |
| 13 | \( 1 \) |
good | 7 | \( 1 + 2iT - 7T^{2} \) |
| 11 | \( 1 - 0.464iT - 11T^{2} \) |
| 17 | \( 1 + 4T + 17T^{2} \) |
| 19 | \( 1 - 0.535iT - 19T^{2} \) |
| 23 | \( 1 - 0.267T + 23T^{2} \) |
| 29 | \( 1 - 3.73T + 29T^{2} \) |
| 31 | \( 1 - 1.73iT - 31T^{2} \) |
| 37 | \( 1 - 1.19iT - 37T^{2} \) |
| 41 | \( 1 + 2iT - 41T^{2} \) |
| 43 | \( 1 + 1.92T + 43T^{2} \) |
| 47 | \( 1 + 10.4iT - 47T^{2} \) |
| 53 | \( 1 + 12.9T + 53T^{2} \) |
| 59 | \( 1 - 1.53iT - 59T^{2} \) |
| 61 | \( 1 - 10.3T + 61T^{2} \) |
| 67 | \( 1 - 4.53iT - 67T^{2} \) |
| 71 | \( 1 - 8.39iT - 71T^{2} \) |
| 73 | \( 1 - 2iT - 73T^{2} \) |
| 79 | \( 1 + 0.0717T + 79T^{2} \) |
| 83 | \( 1 + 4.92iT - 83T^{2} \) |
| 89 | \( 1 - 7.46iT - 89T^{2} \) |
| 97 | \( 1 - 7.46iT - 97T^{2} \) |
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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
−8.335086245811485636338093679666, −7.50354576580451273373249230230, −6.87853040177960279467284463646, −6.50062866834250144056278793920, −5.62500485337379156825898025440, −4.83170884609190271036764818224, −4.17672741057560262241094813117, −3.35434524088864004605204738591, −2.10442867404915427308840673302, −0.799226027606984706254904247300,
0.42419887648281060949837645476, 1.60756357944892055932219947748, 2.47814259297317977755325093201, 3.40559991718207886799801950101, 4.51017453761159592094196942276, 4.85923643996565212245001478707, 5.83955675128917174586473787014, 6.33686905891699146741406733039, 7.33510924233946762497410306890, 8.249316013278847959145133291050