L(s) = 1 | + (−0.809 − 0.587i)2-s + (0.190 − 0.587i)3-s + (0.309 + 0.951i)4-s + (−0.5 + 0.363i)6-s + (0.309 − 0.951i)8-s + (0.5 + 0.363i)9-s + 0.618·12-s + (−0.809 + 0.587i)16-s + (1.30 − 0.951i)17-s + (−0.190 − 0.587i)18-s + (−0.5 + 1.53i)19-s + (−0.5 − 0.363i)24-s + (0.309 − 0.951i)25-s + (0.809 − 0.587i)27-s + 32-s + ⋯ |
L(s) = 1 | + (−0.809 − 0.587i)2-s + (0.190 − 0.587i)3-s + (0.309 + 0.951i)4-s + (−0.5 + 0.363i)6-s + (0.309 − 0.951i)8-s + (0.5 + 0.363i)9-s + 0.618·12-s + (−0.809 + 0.587i)16-s + (1.30 − 0.951i)17-s + (−0.190 − 0.587i)18-s + (−0.5 + 1.53i)19-s + (−0.5 − 0.363i)24-s + (0.309 − 0.951i)25-s + (0.809 − 0.587i)27-s + 32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 968 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.530 + 0.847i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 968 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.530 + 0.847i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7960196297\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7960196297\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.809 + 0.587i)T \) |
| 11 | \( 1 \) |
good | 3 | \( 1 + (-0.190 + 0.587i)T + (-0.809 - 0.587i)T^{2} \) |
| 5 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 7 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 13 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 17 | \( 1 + (-1.30 + 0.951i)T + (0.309 - 0.951i)T^{2} \) |
| 19 | \( 1 + (0.5 - 1.53i)T + (-0.809 - 0.587i)T^{2} \) |
| 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 | \( 1 + (-0.190 + 0.587i)T + (-0.809 - 0.587i)T^{2} \) |
| 43 | \( 1 - 0.618T + T^{2} \) |
| 47 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 53 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 59 | \( 1 + (0.5 + 1.53i)T + (-0.809 + 0.587i)T^{2} \) |
| 61 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 67 | \( 1 + 1.61T + T^{2} \) |
| 71 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 73 | \( 1 + (-0.190 - 0.587i)T + (-0.809 + 0.587i)T^{2} \) |
| 79 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 83 | \( 1 + (0.5 - 0.363i)T + (0.309 - 0.951i)T^{2} \) |
| 89 | \( 1 - 0.618T + T^{2} \) |
| 97 | \( 1 + (0.5 + 0.363i)T + (0.309 + 0.951i)T^{2} \) |
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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
−10.11259501613401412181343544611, −9.373536798114136276684583967655, −8.262963694085976211439185580020, −7.79467691462959932552402181505, −7.03717858337785769635968929019, −6.01394163467952430332041593955, −4.61147687887228281503387112531, −3.47053398743761201590989506766, −2.35681230772135851514002206607, −1.26256999222350926764932468331,
1.37718933754546050902248480483, 2.99591221449679033372099230474, 4.28080413336667920044770606805, 5.23692254945087158657633877156, 6.23188142340669280174983303432, 7.09149062697562172761097032021, 7.88021524417630060264403830909, 8.886083011078873444029899959842, 9.360197644572703001981767176846, 10.24047360438505063388513290703