L(s) = 1 | + (−0.194 − 1.10i)2-s + (1.47 − 1.23i)3-s + (−0.235 + 0.0857i)4-s + (−1.64 − 1.38i)6-s + (0.0348 − 0.0604i)7-s + (−0.418 − 0.725i)8-s + (0.468 − 2.65i)9-s + (0.5 + 0.866i)11-s + (−0.241 + 0.417i)12-s + (−0.766 − 0.642i)13-s + (−0.0733 − 0.0266i)14-s + (−0.909 + 0.763i)16-s − 3.01·18-s + (0.241 + 0.970i)19-s + (−0.0233 − 0.132i)21-s + (0.856 − 0.718i)22-s + ⋯ |
L(s) = 1 | + (−0.194 − 1.10i)2-s + (1.47 − 1.23i)3-s + (−0.235 + 0.0857i)4-s + (−1.64 − 1.38i)6-s + (0.0348 − 0.0604i)7-s + (−0.418 − 0.725i)8-s + (0.468 − 2.65i)9-s + (0.5 + 0.866i)11-s + (−0.241 + 0.417i)12-s + (−0.766 − 0.642i)13-s + (−0.0733 − 0.0266i)14-s + (−0.909 + 0.763i)16-s − 3.01·18-s + (0.241 + 0.970i)19-s + (−0.0233 − 0.132i)21-s + (0.856 − 0.718i)22-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2717 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.996 + 0.0855i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2717 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.996 + 0.0855i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.876448894\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.876448894\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 + (-0.5 - 0.866i)T \) |
| 13 | \( 1 + (0.766 + 0.642i)T \) |
| 19 | \( 1 + (-0.241 - 0.970i)T \) |
good | 2 | \( 1 + (0.194 + 1.10i)T + (-0.939 + 0.342i)T^{2} \) |
| 3 | \( 1 + (-1.47 + 1.23i)T + (0.173 - 0.984i)T^{2} \) |
| 5 | \( 1 + (-0.766 - 0.642i)T^{2} \) |
| 7 | \( 1 + (-0.0348 + 0.0604i)T + (-0.5 - 0.866i)T^{2} \) |
| 17 | \( 1 + (0.939 - 0.342i)T^{2} \) |
| 23 | \( 1 + (1.86 - 0.677i)T + (0.766 - 0.642i)T^{2} \) |
| 29 | \( 1 + (0.939 + 0.342i)T^{2} \) |
| 31 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 + (-1.35 + 1.13i)T + (0.173 - 0.984i)T^{2} \) |
| 43 | \( 1 + (-0.766 - 0.642i)T^{2} \) |
| 47 | \( 1 + (0.939 + 0.342i)T^{2} \) |
| 53 | \( 1 + (-1.35 + 0.492i)T + (0.766 - 0.642i)T^{2} \) |
| 59 | \( 1 + (0.939 - 0.342i)T^{2} \) |
| 61 | \( 1 + (-0.766 + 0.642i)T^{2} \) |
| 67 | \( 1 + (0.939 + 0.342i)T^{2} \) |
| 71 | \( 1 + (-0.766 - 0.642i)T^{2} \) |
| 73 | \( 1 + (-1.23 + 1.04i)T + (0.173 - 0.984i)T^{2} \) |
| 79 | \( 1 + (-0.173 + 0.984i)T^{2} \) |
| 83 | \( 1 + (0.615 - 1.06i)T + (-0.5 - 0.866i)T^{2} \) |
| 89 | \( 1 + (-0.173 - 0.984i)T^{2} \) |
| 97 | \( 1 + (0.939 - 0.342i)T^{2} \) |
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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
−8.760849045368090905057345351580, −7.82190988305360829021262566668, −7.40328235771816354511459882421, −6.64910980948465598262042762910, −5.73414522781198143516039890104, −4.06481959828869647620247591034, −3.52843761725134319587233583473, −2.52217607418903190036913092325, −1.99364605741043858926500135557, −1.11854225696499536659986067879,
2.28901006097455405022257786459, 2.82243939605949383917738065214, 3.98537709382718455452957929333, 4.60313581597504966791649550626, 5.50242616414117599136028149166, 6.47520011682501569567774256200, 7.30402332889262598778853581459, 8.078127166882226598108360620992, 8.562360921398147252494351113937, 9.130101761298355576040516781974