L(s) = 1 | + (−0.115 + 0.356i)3-s + (0.309 + 0.951i)4-s + (−1.56 + 1.13i)5-s + (−0.574 − 1.76i)7-s + (0.695 + 0.505i)9-s + (−0.992 + 0.125i)11-s − 0.374·12-s + (−1.41 − 1.03i)13-s + (−0.224 − 0.690i)15-s + (−0.809 + 0.587i)16-s + (−1.56 − 1.13i)20-s + 0.696·21-s + (0.850 − 2.61i)25-s + (−0.563 + 0.409i)27-s + (1.50 − 1.09i)28-s + ⋯ |
L(s) = 1 | + (−0.115 + 0.356i)3-s + (0.309 + 0.951i)4-s + (−1.56 + 1.13i)5-s + (−0.574 − 1.76i)7-s + (0.695 + 0.505i)9-s + (−0.992 + 0.125i)11-s − 0.374·12-s + (−1.41 − 1.03i)13-s + (−0.224 − 0.690i)15-s + (−0.809 + 0.587i)16-s + (−1.56 − 1.13i)20-s + 0.696·21-s + (0.850 − 2.61i)25-s + (−0.563 + 0.409i)27-s + (1.50 − 1.09i)28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.712 + 0.701i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.712 + 0.701i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.07169405373\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.07169405373\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 + (0.992 - 0.125i)T \) |
| 131 | \( 1 - T \) |
good | 2 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 3 | \( 1 + (0.115 - 0.356i)T + (-0.809 - 0.587i)T^{2} \) |
| 5 | \( 1 + (1.56 - 1.13i)T + (0.309 - 0.951i)T^{2} \) |
| 7 | \( 1 + (0.574 + 1.76i)T + (-0.809 + 0.587i)T^{2} \) |
| 13 | \( 1 + (1.41 + 1.03i)T + (0.309 + 0.951i)T^{2} \) |
| 17 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 19 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 31 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 37 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 41 | \( 1 + (0.263 - 0.809i)T + (-0.809 - 0.587i)T^{2} \) |
| 43 | \( 1 - 0.125T + T^{2} \) |
| 47 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 53 | \( 1 + (0.5 + 0.363i)T + (0.309 + 0.951i)T^{2} \) |
| 59 | \( 1 + (0.393 + 1.21i)T + (-0.809 + 0.587i)T^{2} \) |
| 61 | \( 1 + (1.56 - 1.13i)T + (0.309 - 0.951i)T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 73 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 79 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 83 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 89 | \( 1 + 1.61T + T^{2} \) |
| 97 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
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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
−10.49160434526182870013487303933, −9.789481743408217651888624791408, −7.979582589533618693811447040303, −7.74064700777358857227671538438, −7.25379629443674976525902462117, −6.65985542107144113337333118465, −4.79997720920139552933541322297, −4.17310326602568559515420925175, −3.35548324144925749939161635388, −2.73235762509476184452361648995,
0.05563377149387688622689255639, 1.76504212503075879670923407166, 2.89736579369161970716709029038, 4.36709202975875904729080981900, 5.05757892790510676526717659426, 5.79971771133403065436492706187, 6.86649212756511303655286496237, 7.54298375797258606809128638820, 8.512729156072213607958417391806, 9.290528639862277473262758726542