L(s) = 1 | + (−0.734 − 0.533i)2-s + (−0.0542 − 0.166i)4-s + (0.951 − 0.309i)7-s + (−0.329 + 1.01i)8-s + (0.987 + 0.156i)11-s + (−0.863 − 0.280i)14-s + (0.642 − 0.466i)16-s + (−0.642 − 0.642i)22-s − 1.78i·23-s + (−0.309 + 0.951i)25-s + (−0.103 − 0.142i)28-s + (−0.0966 − 0.297i)29-s + 0.346·32-s + (−0.587 − 1.80i)37-s + 1.61i·43-s + (−0.0274 − 0.173i)44-s + ⋯ |
L(s) = 1 | + (−0.734 − 0.533i)2-s + (−0.0542 − 0.166i)4-s + (0.951 − 0.309i)7-s + (−0.329 + 1.01i)8-s + (0.987 + 0.156i)11-s + (−0.863 − 0.280i)14-s + (0.642 − 0.466i)16-s + (−0.642 − 0.642i)22-s − 1.78i·23-s + (−0.309 + 0.951i)25-s + (−0.103 − 0.142i)28-s + (−0.0966 − 0.297i)29-s + 0.346·32-s + (−0.587 − 1.80i)37-s + 1.61i·43-s + (−0.0274 − 0.173i)44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.402 + 0.915i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.402 + 0.915i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7061868735\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7061868735\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + (-0.951 + 0.309i)T \) |
| 11 | \( 1 + (-0.987 - 0.156i)T \) |
good | 2 | \( 1 + (0.734 + 0.533i)T + (0.309 + 0.951i)T^{2} \) |
| 5 | \( 1 + (0.309 - 0.951i)T^{2} \) |
| 13 | \( 1 + (0.309 + 0.951i)T^{2} \) |
| 17 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 19 | \( 1 + (-0.809 - 0.587i)T^{2} \) |
| 23 | \( 1 + 1.78iT - T^{2} \) |
| 29 | \( 1 + (0.0966 + 0.297i)T + (-0.809 + 0.587i)T^{2} \) |
| 31 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 37 | \( 1 + (0.587 + 1.80i)T + (-0.809 + 0.587i)T^{2} \) |
| 41 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 43 | \( 1 - 1.61iT - T^{2} \) |
| 47 | \( 1 + (-0.809 - 0.587i)T^{2} \) |
| 53 | \( 1 + (-0.183 + 0.253i)T + (-0.309 - 0.951i)T^{2} \) |
| 59 | \( 1 + (-0.809 + 0.587i)T^{2} \) |
| 61 | \( 1 + (0.309 - 0.951i)T^{2} \) |
| 67 | \( 1 + 1.61T + T^{2} \) |
| 71 | \( 1 + (-1.16 - 1.59i)T + (-0.309 + 0.951i)T^{2} \) |
| 73 | \( 1 + (-0.809 + 0.587i)T^{2} \) |
| 79 | \( 1 + (0.690 - 0.951i)T + (-0.309 - 0.951i)T^{2} \) |
| 83 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 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
−10.57349426378619545675900608310, −9.653174689899536106897026409875, −8.905522486581032259786156313159, −8.196570808172035148461442265304, −7.18288744374796686724519578318, −6.04606971751183770991396506099, −4.98461130327721379562703744967, −4.00680523639470954762419679314, −2.35405747009839158525114193277, −1.23470822574348972553277143840,
1.56068324131744474829727750197, 3.33878064382717655120915340626, 4.38839005943541075756168089498, 5.61202361112755444232928647323, 6.64546497063641141844514118203, 7.50340724305552595658435391941, 8.284580128500657370635364216275, 8.949765729037654034744893271477, 9.691169460784368002436702431350, 10.69672024028364157074166705647