L(s) = 1 | + (1.70 − 0.984i)7-s + (−0.266 − 1.50i)13-s + (0.5 + 0.866i)19-s + (−0.173 − 0.984i)25-s + (−0.592 + 0.342i)31-s − 1.87·37-s + (−0.439 + 0.524i)43-s + (1.43 − 2.49i)49-s + (1.17 − 0.984i)61-s + (−0.673 + 1.85i)67-s + (−0.266 + 1.50i)73-s + (1.93 + 0.342i)79-s + (−1.93 − 2.31i)91-s + (−0.939 + 0.342i)97-s + (1.11 + 0.642i)103-s + ⋯ |
L(s) = 1 | + (1.70 − 0.984i)7-s + (−0.266 − 1.50i)13-s + (0.5 + 0.866i)19-s + (−0.173 − 0.984i)25-s + (−0.592 + 0.342i)31-s − 1.87·37-s + (−0.439 + 0.524i)43-s + (1.43 − 2.49i)49-s + (1.17 − 0.984i)61-s + (−0.673 + 1.85i)67-s + (−0.266 + 1.50i)73-s + (1.93 + 0.342i)79-s + (−1.93 − 2.31i)91-s + (−0.939 + 0.342i)97-s + (1.11 + 0.642i)103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.672 + 0.740i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.672 + 0.740i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.445135943\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.445135943\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 + (-0.5 - 0.866i)T \) |
good | 5 | \( 1 + (0.173 + 0.984i)T^{2} \) |
| 7 | \( 1 + (-1.70 + 0.984i)T + (0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 + (0.266 + 1.50i)T + (-0.939 + 0.342i)T^{2} \) |
| 17 | \( 1 + (0.766 - 0.642i)T^{2} \) |
| 23 | \( 1 + (-0.173 + 0.984i)T^{2} \) |
| 29 | \( 1 + (0.766 + 0.642i)T^{2} \) |
| 31 | \( 1 + (0.592 - 0.342i)T + (0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + 1.87T + T^{2} \) |
| 41 | \( 1 + (-0.939 - 0.342i)T^{2} \) |
| 43 | \( 1 + (0.439 - 0.524i)T + (-0.173 - 0.984i)T^{2} \) |
| 47 | \( 1 + (-0.766 - 0.642i)T^{2} \) |
| 53 | \( 1 + (0.173 - 0.984i)T^{2} \) |
| 59 | \( 1 + (-0.766 + 0.642i)T^{2} \) |
| 61 | \( 1 + (-1.17 + 0.984i)T + (0.173 - 0.984i)T^{2} \) |
| 67 | \( 1 + (0.673 - 1.85i)T + (-0.766 - 0.642i)T^{2} \) |
| 71 | \( 1 + (-0.173 - 0.984i)T^{2} \) |
| 73 | \( 1 + (0.266 - 1.50i)T + (-0.939 - 0.342i)T^{2} \) |
| 79 | \( 1 + (-1.93 - 0.342i)T + (0.939 + 0.342i)T^{2} \) |
| 83 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 89 | \( 1 + (-0.939 + 0.342i)T^{2} \) |
| 97 | \( 1 + (0.939 - 0.342i)T + (0.766 - 0.642i)T^{2} \) |
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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
−8.578842431067119737246515234921, −8.123532973846039146538908869811, −7.55790541402751341260690301822, −6.81653154910947266112633030739, −5.53014926531502327123002378274, −5.12627693218936796123160523509, −4.17792939747755675968107813779, −3.35347556209531539868810905284, −2.04753858091819288430427177756, −1.02134320580787487600268775313,
1.65383534467245632856282335117, 2.18261228895643057486229214107, 3.50761405254607011434148827038, 4.67026880240693195162183169745, 5.06543754003801735963696710384, 5.90762812265530115618272991299, 7.01456553909952176897685990955, 7.54138612005558656251061701992, 8.537354226016841855303327643472, 8.971732591742995713266219319200