L(s) = 1 | + (0.968 − 2.65i)2-s + (−6.12 − 5.14i)4-s + 12.6i·5-s − 7i·7-s + (−19.6 + 11.2i)8-s + (33.6 + 12.2i)10-s + 54.6·11-s + 20.4·13-s + (−18.6 − 6.77i)14-s + (11.0 + 63.0i)16-s − 134. i·17-s − 9.73i·19-s + (65.1 − 77.4i)20-s + (52.9 − 145. i)22-s + 161.·23-s + ⋯ |
L(s) = 1 | + (0.342 − 0.939i)2-s + (−0.765 − 0.643i)4-s + 1.13i·5-s − 0.377i·7-s + (−0.866 + 0.498i)8-s + (1.06 + 0.387i)10-s + 1.49·11-s + 0.435·13-s + (−0.355 − 0.129i)14-s + (0.171 + 0.985i)16-s − 1.92i·17-s − 0.117i·19-s + (0.728 − 0.866i)20-s + (0.513 − 1.40i)22-s + 1.46·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.253 + 0.967i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.253 + 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(2.103471759\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.103471759\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.968 + 2.65i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + 7iT \) |
good | 5 | \( 1 - 12.6iT - 125T^{2} \) |
| 11 | \( 1 - 54.6T + 1.33e3T^{2} \) |
| 13 | \( 1 - 20.4T + 2.19e3T^{2} \) |
| 17 | \( 1 + 134. iT - 4.91e3T^{2} \) |
| 19 | \( 1 + 9.73iT - 6.85e3T^{2} \) |
| 23 | \( 1 - 161.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 216. iT - 2.43e4T^{2} \) |
| 31 | \( 1 - 240. iT - 2.97e4T^{2} \) |
| 37 | \( 1 - 72.2T + 5.06e4T^{2} \) |
| 41 | \( 1 - 284. iT - 6.89e4T^{2} \) |
| 43 | \( 1 + 402. iT - 7.95e4T^{2} \) |
| 47 | \( 1 - 286.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 236. iT - 1.48e5T^{2} \) |
| 59 | \( 1 + 445.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 230.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 697. iT - 3.00e5T^{2} \) |
| 71 | \( 1 - 786.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 400.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 119. iT - 4.93e5T^{2} \) |
| 83 | \( 1 - 1.30e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 557. iT - 7.04e5T^{2} \) |
| 97 | \( 1 - 149.T + 9.12e5T^{2} \) |
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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
−11.40004638680635275909141753094, −10.69237818956246866387967214841, −9.643262466622897840125175033699, −8.890115932286247060616615423148, −7.18290000153863825255949204965, −6.38881960888264959091990453745, −4.89339312517970968007868305997, −3.65587381375519703259023137860, −2.69836070897661500238188503671, −0.971149896330171470788489566358,
1.23227077816894202597298529877, 3.65087609304722677319649123792, 4.60233322187694249108875478443, 5.77562052682808028914254629948, 6.58894078820172479297972508581, 7.958237916403170280107482609597, 8.913264270513893359590618557457, 9.246381532043135222349010015445, 10.96406548059888283755995767926, 12.28839948002654029313464936563