L(s) = 1 | + (−0.888 + 0.458i)3-s + (0.654 − 0.755i)4-s + (−0.723 + 0.690i)7-s + (0.580 − 0.814i)9-s + (−0.235 + 0.971i)12-s + (0.195 − 0.807i)13-s + (−0.142 − 0.989i)16-s + (1.81 + 0.829i)19-s + (0.327 − 0.945i)21-s + (0.235 − 0.971i)25-s + (−0.142 + 0.989i)27-s + (0.0475 + 0.998i)28-s + (−0.273 − 0.0801i)31-s + (−0.235 − 0.971i)36-s + 1.44·37-s + ⋯ |
L(s) = 1 | + (−0.888 + 0.458i)3-s + (0.654 − 0.755i)4-s + (−0.723 + 0.690i)7-s + (0.580 − 0.814i)9-s + (−0.235 + 0.971i)12-s + (0.195 − 0.807i)13-s + (−0.142 − 0.989i)16-s + (1.81 + 0.829i)19-s + (0.327 − 0.945i)21-s + (0.235 − 0.971i)25-s + (−0.142 + 0.989i)27-s + (0.0475 + 0.998i)28-s + (−0.273 − 0.0801i)31-s + (−0.235 − 0.971i)36-s + 1.44·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1407 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.952 + 0.303i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1407 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.952 + 0.303i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9223052407\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9223052407\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.888 - 0.458i)T \) |
| 7 | \( 1 + (0.723 - 0.690i)T \) |
| 67 | \( 1 + (0.723 + 0.690i)T \) |
good | 2 | \( 1 + (-0.654 + 0.755i)T^{2} \) |
| 5 | \( 1 + (-0.235 + 0.971i)T^{2} \) |
| 11 | \( 1 + (-0.959 + 0.281i)T^{2} \) |
| 13 | \( 1 + (-0.195 + 0.807i)T + (-0.888 - 0.458i)T^{2} \) |
| 17 | \( 1 + (-0.786 - 0.618i)T^{2} \) |
| 19 | \( 1 + (-1.81 - 0.829i)T + (0.654 + 0.755i)T^{2} \) |
| 23 | \( 1 + (-0.415 - 0.909i)T^{2} \) |
| 29 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 + (0.273 + 0.0801i)T + (0.841 + 0.540i)T^{2} \) |
| 37 | \( 1 - 1.44T + T^{2} \) |
| 41 | \( 1 + (0.786 + 0.618i)T^{2} \) |
| 43 | \( 1 + (-0.425 + 0.368i)T + (0.142 - 0.989i)T^{2} \) |
| 47 | \( 1 + (0.580 - 0.814i)T^{2} \) |
| 53 | \( 1 + (-0.786 + 0.618i)T^{2} \) |
| 59 | \( 1 + (0.0475 - 0.998i)T^{2} \) |
| 61 | \( 1 + (-0.0135 + 0.0941i)T + (-0.959 - 0.281i)T^{2} \) |
| 71 | \( 1 + (-0.928 - 0.371i)T^{2} \) |
| 73 | \( 1 + (-0.459 - 1.14i)T + (-0.723 + 0.690i)T^{2} \) |
| 79 | \( 1 + (-1.34 - 0.325i)T + (0.888 + 0.458i)T^{2} \) |
| 83 | \( 1 + (0.723 + 0.690i)T^{2} \) |
| 89 | \( 1 + (0.580 + 0.814i)T^{2} \) |
| 97 | \( 1 + (0.888 - 1.53i)T + (-0.5 - 0.866i)T^{2} \) |
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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
−9.754487437243484465925337064452, −9.368245486672109599768077297906, −8.012675252676469770574489190053, −7.04192452362613574279960044990, −6.15843258914207495786034125333, −5.70128327775226255416061220235, −5.00668824223363805914146145283, −3.63830507111902730987419856274, −2.62620327795568007119776736607, −1.02355783586509127799310358180,
1.28466571490235707264091840788, 2.72293803449968649069543965279, 3.72866054890062653324148181357, 4.76060412408262166867695429222, 5.89071633653734977834447742447, 6.67339196935065081307738978649, 7.29272666704360027657821651874, 7.74884883877090879549440667940, 9.090278440268188454061902502936, 9.817601183151601120579005454131