L(s) = 1 | + (−0.930 + 0.365i)2-s + (0.733 − 0.680i)4-s + (−0.246 + 0.167i)5-s + (0.826 + 0.563i)7-s + (−0.433 + 0.900i)8-s + (0.167 − 0.246i)10-s + (−0.218 + 1.44i)11-s + (−0.974 − 0.222i)14-s + (0.0747 − 0.997i)16-s + (−0.0663 + 0.290i)20-s + (−0.326 − 1.42i)22-s + (−0.332 + 0.848i)25-s + (0.988 − 0.149i)28-s + (0.712 + 0.162i)29-s + (−1.72 + 0.997i)31-s + (0.294 + 0.955i)32-s + ⋯ |
L(s) = 1 | + (−0.930 + 0.365i)2-s + (0.733 − 0.680i)4-s + (−0.246 + 0.167i)5-s + (0.826 + 0.563i)7-s + (−0.433 + 0.900i)8-s + (0.167 − 0.246i)10-s + (−0.218 + 1.44i)11-s + (−0.974 − 0.222i)14-s + (0.0747 − 0.997i)16-s + (−0.0663 + 0.290i)20-s + (−0.326 − 1.42i)22-s + (−0.332 + 0.848i)25-s + (0.988 − 0.149i)28-s + (0.712 + 0.162i)29-s + (−1.72 + 0.997i)31-s + (0.294 + 0.955i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.518 - 0.855i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.518 - 0.855i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7022305454\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7022305454\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.930 - 0.365i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-0.826 - 0.563i)T \) |
good | 5 | \( 1 + (0.246 - 0.167i)T + (0.365 - 0.930i)T^{2} \) |
| 11 | \( 1 + (0.218 - 1.44i)T + (-0.955 - 0.294i)T^{2} \) |
| 13 | \( 1 + (-0.222 - 0.974i)T^{2} \) |
| 17 | \( 1 + (-0.826 + 0.563i)T^{2} \) |
| 19 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 + (0.826 + 0.563i)T^{2} \) |
| 29 | \( 1 + (-0.712 - 0.162i)T + (0.900 + 0.433i)T^{2} \) |
| 31 | \( 1 + (1.72 - 0.997i)T + (0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + (-0.0747 - 0.997i)T^{2} \) |
| 41 | \( 1 + (-0.623 - 0.781i)T^{2} \) |
| 43 | \( 1 + (-0.623 + 0.781i)T^{2} \) |
| 47 | \( 1 + (0.733 - 0.680i)T^{2} \) |
| 53 | \( 1 + (0.680 + 0.733i)T + (-0.0747 + 0.997i)T^{2} \) |
| 59 | \( 1 + (1.43 + 0.975i)T + (0.365 + 0.930i)T^{2} \) |
| 61 | \( 1 + (0.0747 + 0.997i)T^{2} \) |
| 67 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 + (-0.900 + 0.433i)T^{2} \) |
| 73 | \( 1 + (0.807 + 0.317i)T + (0.733 + 0.680i)T^{2} \) |
| 79 | \( 1 + (-0.826 + 1.43i)T + (-0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 + (-1.16 - 1.45i)T + (-0.222 + 0.974i)T^{2} \) |
| 89 | \( 1 + (-0.955 + 0.294i)T^{2} \) |
| 97 | \( 1 - 0.589iT - T^{2} \) |
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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
−9.067890138095837356959723776300, −8.166327893276351526835102855561, −7.57927788447719265587710465047, −7.05511323297420326431689109077, −6.18324519751974035272498284990, −5.19768922526796257647685769969, −4.76250691891918861348296793220, −3.37751903222506865047405308878, −2.18412842740485329509249718573, −1.56680934053516157428729409084,
0.55892190719950638605345557258, 1.67739150551778572664969231956, 2.78791680394670560968213178627, 3.69989056074520722769045800394, 4.46792795331858971837543791131, 5.64992274500464997081011295747, 6.35652657330505416643805441444, 7.40695373045765272049076221881, 7.87035203546041289329346223783, 8.507251230403708685347956144261