L(s) = 1 | + (−0.613 − 1.88i)3-s + (0.309 − 0.951i)4-s + (1.03 + 0.749i)5-s + (0.598 − 1.84i)7-s + (−2.37 + 1.72i)9-s + (−0.425 + 0.904i)11-s − 1.98·12-s + (0.303 − 0.220i)13-s + (0.781 − 2.40i)15-s + (−0.809 − 0.587i)16-s + (1.03 − 0.749i)20-s − 3.84·21-s + (0.193 + 0.594i)25-s + (3.10 + 2.25i)27-s + (−1.56 − 1.13i)28-s + ⋯ |
L(s) = 1 | + (−0.613 − 1.88i)3-s + (0.309 − 0.951i)4-s + (1.03 + 0.749i)5-s + (0.598 − 1.84i)7-s + (−2.37 + 1.72i)9-s + (−0.425 + 0.904i)11-s − 1.98·12-s + (0.303 − 0.220i)13-s + (0.781 − 2.40i)15-s + (−0.809 − 0.587i)16-s + (1.03 − 0.749i)20-s − 3.84·21-s + (0.193 + 0.594i)25-s + (3.10 + 2.25i)27-s + (−1.56 − 1.13i)28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.887 + 0.460i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.887 + 0.460i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.111074086\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.111074086\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 + (0.425 - 0.904i)T \) |
| 131 | \( 1 - T \) |
good | 2 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 3 | \( 1 + (0.613 + 1.88i)T + (-0.809 + 0.587i)T^{2} \) |
| 5 | \( 1 + (-1.03 - 0.749i)T + (0.309 + 0.951i)T^{2} \) |
| 7 | \( 1 + (-0.598 + 1.84i)T + (-0.809 - 0.587i)T^{2} \) |
| 13 | \( 1 + (-0.303 + 0.220i)T + (0.309 - 0.951i)T^{2} \) |
| 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 | \( 1 + (-0.450 - 1.38i)T + (-0.809 + 0.587i)T^{2} \) |
| 43 | \( 1 - 1.07T + T^{2} \) |
| 47 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 53 | \( 1 + (0.5 - 0.363i)T + (0.309 - 0.951i)T^{2} \) |
| 59 | \( 1 + (-0.0388 + 0.119i)T + (-0.809 - 0.587i)T^{2} \) |
| 61 | \( 1 + (-1.03 - 0.749i)T + (0.309 + 0.951i)T^{2} \) |
| 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
−9.710182705954584309205766246055, −8.161533452882893072531014436272, −7.36293984421791203779776273709, −6.96850885805695072825916070077, −6.29552528913398755438474902777, −5.58750834094308254453725581054, −4.63591013335201917488151613379, −2.66392466326180335927685706358, −1.77950587063949820314605008006, −1.03024471473904774924360263067,
2.29300372664594227228657079201, 3.18257998736867780663865013726, 4.27496882544134104259520021538, 5.25002080807157209692250930165, 5.62084046717936386133323286614, 6.29605222750459110748903808287, 8.175980446973584879937408003373, 8.849242634853497799885374254208, 9.059051040029955153473914798733, 9.914929781500349179600604120876