L(s) = 1 | + (1.19 + 0.756i)2-s − i·3-s + (0.856 + 1.80i)4-s − 4.10i·5-s + (0.756 − 1.19i)6-s + 7-s + (−0.343 + 2.80i)8-s − 9-s + (3.10 − 4.90i)10-s + 2.67i·11-s + (1.80 − 0.856i)12-s + 3.02i·13-s + (1.19 + 0.756i)14-s − 4.10·15-s + (−2.53 + 3.09i)16-s − 5.12·17-s + ⋯ |
L(s) = 1 | + (0.845 + 0.534i)2-s − 0.577i·3-s + (0.428 + 0.903i)4-s − 1.83i·5-s + (0.308 − 0.487i)6-s + 0.377·7-s + (−0.121 + 0.992i)8-s − 0.333·9-s + (0.981 − 1.55i)10-s + 0.807i·11-s + (0.521 − 0.247i)12-s + 0.838i·13-s + (0.319 + 0.202i)14-s − 1.05·15-s + (−0.633 + 0.773i)16-s − 1.24·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 + 0.121i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.992 + 0.121i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.75768 - 0.107192i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.75768 - 0.107192i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.19 - 0.756i)T \) |
| 3 | \( 1 + iT \) |
| 7 | \( 1 - T \) |
good | 5 | \( 1 + 4.10iT - 5T^{2} \) |
| 11 | \( 1 - 2.67iT - 11T^{2} \) |
| 13 | \( 1 - 3.02iT - 13T^{2} \) |
| 17 | \( 1 + 5.12T + 17T^{2} \) |
| 19 | \( 1 - 2.78iT - 19T^{2} \) |
| 23 | \( 1 - 7.12T + 23T^{2} \) |
| 29 | \( 1 + 8.83iT - 29T^{2} \) |
| 31 | \( 1 + 1.42T + 31T^{2} \) |
| 37 | \( 1 - 1.42iT - 37T^{2} \) |
| 41 | \( 1 - 5.12T + 41T^{2} \) |
| 43 | \( 1 - 2.39iT - 43T^{2} \) |
| 47 | \( 1 + 9.56T + 47T^{2} \) |
| 53 | \( 1 + 2.78iT - 53T^{2} \) |
| 59 | \( 1 - 4iT - 59T^{2} \) |
| 61 | \( 1 + 5.17iT - 61T^{2} \) |
| 67 | \( 1 + 0.244iT - 67T^{2} \) |
| 71 | \( 1 + 4.27T + 71T^{2} \) |
| 73 | \( 1 + 4.15T + 73T^{2} \) |
| 79 | \( 1 - 6.25T + 79T^{2} \) |
| 83 | \( 1 + 9.35iT - 83T^{2} \) |
| 89 | \( 1 - 11.2T + 89T^{2} \) |
| 97 | \( 1 - 6.69T + 97T^{2} \) |
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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
−13.00932823495184399991630996761, −12.08842355729510033553162983564, −11.38515124320642213491992681606, −9.328340791274012825436172790513, −8.488778980249549280099455526162, −7.53166793441920669013715216149, −6.28276572866599524089765463043, −4.97824684458435886029360903309, −4.30665514206911784230827900037, −1.91184017365368862940752300597,
2.68275868538750246289004239567, 3.50451381756908344567578205162, 5.04967786297854292469162408271, 6.29628129727940572141124963117, 7.23568005325900047598719943798, 9.013402375454997787180091614050, 10.38676571614856005567172501133, 10.98451095457600095203818093318, 11.34987956860681282065895596181, 12.96469110797673485585997647486