L(s) = 1 | − 2-s − 3.24·3-s + 4-s + 2.20·5-s + 3.24·6-s + 3.82·7-s − 8-s + 7.54·9-s − 2.20·10-s + 2.47·11-s − 3.24·12-s + 2.31·13-s − 3.82·14-s − 7.15·15-s + 16-s − 2.44·17-s − 7.54·18-s + 19-s + 2.20·20-s − 12.4·21-s − 2.47·22-s − 1.05·23-s + 3.24·24-s − 0.141·25-s − 2.31·26-s − 14.7·27-s + 3.82·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 1.87·3-s + 0.5·4-s + 0.985·5-s + 1.32·6-s + 1.44·7-s − 0.353·8-s + 2.51·9-s − 0.697·10-s + 0.746·11-s − 0.937·12-s + 0.640·13-s − 1.02·14-s − 1.84·15-s + 0.250·16-s − 0.593·17-s − 1.77·18-s + 0.229·19-s + 0.492·20-s − 2.70·21-s − 0.528·22-s − 0.219·23-s + 0.662·24-s − 0.0282·25-s − 0.453·26-s − 2.84·27-s + 0.722·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.167059517\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.167059517\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| 211 | \( 1 + T \) |
good | 3 | \( 1 + 3.24T + 3T^{2} \) |
| 5 | \( 1 - 2.20T + 5T^{2} \) |
| 7 | \( 1 - 3.82T + 7T^{2} \) |
| 11 | \( 1 - 2.47T + 11T^{2} \) |
| 13 | \( 1 - 2.31T + 13T^{2} \) |
| 17 | \( 1 + 2.44T + 17T^{2} \) |
| 23 | \( 1 + 1.05T + 23T^{2} \) |
| 29 | \( 1 + 8.02T + 29T^{2} \) |
| 31 | \( 1 + 3.43T + 31T^{2} \) |
| 37 | \( 1 - 8.66T + 37T^{2} \) |
| 41 | \( 1 + 10.6T + 41T^{2} \) |
| 43 | \( 1 - 8.90T + 43T^{2} \) |
| 47 | \( 1 + 8.78T + 47T^{2} \) |
| 53 | \( 1 + 10.9T + 53T^{2} \) |
| 59 | \( 1 - 0.954T + 59T^{2} \) |
| 61 | \( 1 - 5.86T + 61T^{2} \) |
| 67 | \( 1 - 1.10T + 67T^{2} \) |
| 71 | \( 1 - 3.89T + 71T^{2} \) |
| 73 | \( 1 - 8.71T + 73T^{2} \) |
| 79 | \( 1 - 12.3T + 79T^{2} \) |
| 83 | \( 1 + 15.6T + 83T^{2} \) |
| 89 | \( 1 - 7.32T + 89T^{2} \) |
| 97 | \( 1 - 8.66T + 97T^{2} \) |
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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
−7.72456135419797032447080414304, −7.02401359646759156496663567503, −6.30115632784640160646376056698, −5.88716251767357037075047753948, −5.22822543043036501640443714524, −4.59895720461673572489828265567, −3.73448144868354291363152849615, −1.90051212426071948095166457410, −1.67398017105457304672918748799, −0.69912204280996832386029787483,
0.69912204280996832386029787483, 1.67398017105457304672918748799, 1.90051212426071948095166457410, 3.73448144868354291363152849615, 4.59895720461673572489828265567, 5.22822543043036501640443714524, 5.88716251767357037075047753948, 6.30115632784640160646376056698, 7.02401359646759156496663567503, 7.72456135419797032447080414304