L(s) = 1 | + (0.888 + 0.458i)2-s + (1.11 − 0.326i)3-s + (0.580 + 0.814i)4-s + (−0.654 − 0.755i)5-s + (1.13 + 0.219i)6-s + (−0.0688 − 1.44i)7-s + (0.142 + 0.989i)8-s + (0.290 − 0.186i)9-s + (−0.235 − 0.971i)10-s + (0.911 + 0.717i)12-s + (0.601 − 1.31i)14-s + (−0.975 − 0.627i)15-s + (−0.327 + 0.945i)16-s + (0.344 − 0.0328i)18-s + (0.235 − 0.971i)20-s + (−0.549 − 1.58i)21-s + ⋯ |
L(s) = 1 | + (0.888 + 0.458i)2-s + (1.11 − 0.326i)3-s + (0.580 + 0.814i)4-s + (−0.654 − 0.755i)5-s + (1.13 + 0.219i)6-s + (−0.0688 − 1.44i)7-s + (0.142 + 0.989i)8-s + (0.290 − 0.186i)9-s + (−0.235 − 0.971i)10-s + (0.911 + 0.717i)12-s + (0.601 − 1.31i)14-s + (−0.975 − 0.627i)15-s + (−0.327 + 0.945i)16-s + (0.344 − 0.0328i)18-s + (0.235 − 0.971i)20-s + (−0.549 − 1.58i)21-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1340 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 + 0.109i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1340 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 + 0.109i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.214538791\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.214538791\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.888 - 0.458i)T \) |
| 5 | \( 1 + (0.654 + 0.755i)T \) |
| 67 | \( 1 + (-0.995 + 0.0950i)T \) |
good | 3 | \( 1 + (-1.11 + 0.326i)T + (0.841 - 0.540i)T^{2} \) |
| 7 | \( 1 + (0.0688 + 1.44i)T + (-0.995 + 0.0950i)T^{2} \) |
| 11 | \( 1 + (-0.928 + 0.371i)T^{2} \) |
| 13 | \( 1 + (-0.723 + 0.690i)T^{2} \) |
| 17 | \( 1 + (0.327 + 0.945i)T^{2} \) |
| 19 | \( 1 + (0.995 + 0.0950i)T^{2} \) |
| 23 | \( 1 + (-0.723 - 0.690i)T + (0.0475 + 0.998i)T^{2} \) |
| 29 | \( 1 + (0.235 - 0.408i)T + (-0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 + (-0.723 - 0.690i)T^{2} \) |
| 37 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 41 | \( 1 + (1.84 + 0.176i)T + (0.981 + 0.189i)T^{2} \) |
| 43 | \( 1 + (-0.827 - 1.81i)T + (-0.654 + 0.755i)T^{2} \) |
| 47 | \( 1 + (-0.370 + 1.52i)T + (-0.888 - 0.458i)T^{2} \) |
| 53 | \( 1 + (0.654 + 0.755i)T^{2} \) |
| 59 | \( 1 + (0.959 - 0.281i)T^{2} \) |
| 61 | \( 1 + (1.28 + 0.247i)T + (0.928 + 0.371i)T^{2} \) |
| 71 | \( 1 + (0.327 - 0.945i)T^{2} \) |
| 73 | \( 1 + (-0.928 - 0.371i)T^{2} \) |
| 79 | \( 1 + (-0.235 - 0.971i)T^{2} \) |
| 83 | \( 1 + (-0.0311 + 0.0899i)T + (-0.786 - 0.618i)T^{2} \) |
| 89 | \( 1 + (1.61 + 0.474i)T + (0.841 + 0.540i)T^{2} \) |
| 97 | \( 1 + (0.5 - 0.866i)T^{2} \) |
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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
−9.564812782732498258780748686832, −8.637307887161526281086042304044, −8.017100080730622389908843445291, −7.37068074477440693971945520659, −6.84207421002096800874234685245, −5.44102911096861073370887706515, −4.54824197063308724116452737484, −3.72556013731742324802023967946, −3.10194850920351601932510206718, −1.57539396229218537181275368488,
2.19322921928579838890128005329, 2.83202719308366964475692841078, 3.51924547177282568908754351302, 4.47685714803072346271672724539, 5.56926732096486874234992787601, 6.40724541062170139696439162339, 7.35204921248360740426960219592, 8.378532611484218606936770788093, 9.028389302828196650773215108878, 9.838889572136664867408449906693