L(s) = 1 | + (−1.33 + 0.474i)2-s + (1.54 − 1.26i)4-s − 1.58·5-s + (−2.37 + 1.15i)7-s + (−1.46 + 2.42i)8-s + (2.10 − 0.750i)10-s + 2.26·11-s + 0.548·13-s + (2.62 − 2.66i)14-s + (0.798 − 3.91i)16-s − 0.433i·17-s − 6.02i·19-s + (−2.44 + 1.99i)20-s + (−3.01 + 1.07i)22-s − 8.24i·23-s + ⋯ |
L(s) = 1 | + (−0.941 + 0.335i)2-s + (0.774 − 0.632i)4-s − 0.706·5-s + (−0.899 + 0.436i)7-s + (−0.517 + 0.855i)8-s + (0.665 − 0.237i)10-s + 0.681·11-s + 0.152·13-s + (0.700 − 0.713i)14-s + (0.199 − 0.979i)16-s − 0.105i·17-s − 1.38i·19-s + (−0.547 + 0.447i)20-s + (−0.642 + 0.228i)22-s − 1.71i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0911 + 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0911 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.273447 - 0.299622i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.273447 - 0.299622i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.33 - 0.474i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (2.37 - 1.15i)T \) |
good | 5 | \( 1 + 1.58T + 5T^{2} \) |
| 11 | \( 1 - 2.26T + 11T^{2} \) |
| 13 | \( 1 - 0.548T + 13T^{2} \) |
| 17 | \( 1 + 0.433iT - 17T^{2} \) |
| 19 | \( 1 + 6.02iT - 19T^{2} \) |
| 23 | \( 1 + 8.24iT - 23T^{2} \) |
| 29 | \( 1 - 0.548iT - 29T^{2} \) |
| 31 | \( 1 + 7.50T + 31T^{2} \) |
| 37 | \( 1 + 4.21iT - 37T^{2} \) |
| 41 | \( 1 + 7.09iT - 41T^{2} \) |
| 43 | \( 1 + 1.82T + 43T^{2} \) |
| 47 | \( 1 + 11.5T + 47T^{2} \) |
| 53 | \( 1 - 3.71iT - 53T^{2} \) |
| 59 | \( 1 + 11.5iT - 59T^{2} \) |
| 61 | \( 1 - 12.1T + 61T^{2} \) |
| 67 | \( 1 - 9.35T + 67T^{2} \) |
| 71 | \( 1 + 1.27iT - 71T^{2} \) |
| 73 | \( 1 - 0.867iT - 73T^{2} \) |
| 79 | \( 1 + 10.3iT - 79T^{2} \) |
| 83 | \( 1 - 6.13iT - 83T^{2} \) |
| 89 | \( 1 - 7.95iT - 89T^{2} \) |
| 97 | \( 1 - 19.1iT - 97T^{2} \) |
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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
−10.64246496881489929644517563395, −9.530326884059829712184153941896, −8.966974677001034224643945975289, −8.114722757194669059043159416513, −6.96220667628299300895983236168, −6.46980692302056316948920992087, −5.20962366243742187966970791392, −3.70058007190576117682845044631, −2.35513972243507150185257457460, −0.33194572095947720914511846655,
1.51110531344219540332982692395, 3.37011598431310275921246860855, 3.89348658680321771848724322685, 5.86185574518273226440246831836, 6.86505648782874494977208272215, 7.66466229643005944254557644416, 8.473289462967331423408819572442, 9.633314654594447027295748861476, 9.981044125946210912265269450092, 11.23279240705269328556536991948