L(s) = 1 | − 5-s + (2.23 + 1.41i)7-s + 1.41i·11-s + 1.74i·13-s − 4.47·17-s + 1.74i·19-s + 3.16i·23-s + 25-s − 2.08i·29-s + 4.57i·31-s + (−2.23 − 1.41i)35-s + 1.52·37-s − 0.472·41-s − 0.472·43-s − 2.47·47-s + ⋯ |
L(s) = 1 | − 0.447·5-s + (0.845 + 0.534i)7-s + 0.426i·11-s + 0.484i·13-s − 1.08·17-s + 0.401i·19-s + 0.659i·23-s + 0.200·25-s − 0.386i·29-s + 0.821i·31-s + (−0.377 − 0.239i)35-s + 0.251·37-s − 0.0737·41-s − 0.0720·43-s − 0.360·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0515 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0515 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.289511417\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.289511417\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + (-2.23 - 1.41i)T \) |
good | 11 | \( 1 - 1.41iT - 11T^{2} \) |
| 13 | \( 1 - 1.74iT - 13T^{2} \) |
| 17 | \( 1 + 4.47T + 17T^{2} \) |
| 19 | \( 1 - 1.74iT - 19T^{2} \) |
| 23 | \( 1 - 3.16iT - 23T^{2} \) |
| 29 | \( 1 + 2.08iT - 29T^{2} \) |
| 31 | \( 1 - 4.57iT - 31T^{2} \) |
| 37 | \( 1 - 1.52T + 37T^{2} \) |
| 41 | \( 1 + 0.472T + 41T^{2} \) |
| 43 | \( 1 + 0.472T + 43T^{2} \) |
| 47 | \( 1 + 2.47T + 47T^{2} \) |
| 53 | \( 1 - 8.81iT - 53T^{2} \) |
| 59 | \( 1 + 1.52T + 59T^{2} \) |
| 61 | \( 1 - 3.49iT - 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 - 7.73iT - 71T^{2} \) |
| 73 | \( 1 - 16.5iT - 73T^{2} \) |
| 79 | \( 1 - 8.94T + 79T^{2} \) |
| 83 | \( 1 - 5.52T + 83T^{2} \) |
| 89 | \( 1 + 10.9T + 89T^{2} \) |
| 97 | \( 1 + 0.412iT - 97T^{2} \) |
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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
−9.825578025298321666963985197614, −8.971937752997388171371923620373, −8.329625450514144043657538935725, −7.49540265833210900154735670266, −6.68820084606764138739898179121, −5.63636199517904002775308532236, −4.71772800988564053240207090529, −3.98661509134484489436883332407, −2.61693556462327820932714466230, −1.54152698799098190199158213886,
0.55204498418471840298526922929, 2.07457981900935902566199092608, 3.35124525566485841973865349282, 4.38138444898886458203700803940, 5.04882602754064769915934492739, 6.23186112287728978611856607465, 7.08471910616495079392157428603, 7.950224602576063080022371967282, 8.508559508465249325210021117206, 9.406166494021649442320541379790