L(s) = 1 | − 1.70·2-s + 3.00·3-s + 0.914·4-s + 2.58·5-s − 5.12·6-s − 2.99·7-s + 1.85·8-s + 6.02·9-s − 4.40·10-s + 11-s + 2.74·12-s + 2.82·13-s + 5.11·14-s + 7.75·15-s − 4.99·16-s − 17-s − 10.2·18-s − 4.52·19-s + 2.36·20-s − 8.99·21-s − 1.70·22-s + 8.03·23-s + 5.56·24-s + 1.66·25-s − 4.81·26-s + 9.08·27-s − 2.73·28-s + ⋯ |
L(s) = 1 | − 1.20·2-s + 1.73·3-s + 0.457·4-s + 1.15·5-s − 2.09·6-s − 1.13·7-s + 0.655·8-s + 2.00·9-s − 1.39·10-s + 0.301·11-s + 0.792·12-s + 0.782·13-s + 1.36·14-s + 2.00·15-s − 1.24·16-s − 0.242·17-s − 2.42·18-s − 1.03·19-s + 0.527·20-s − 1.96·21-s − 0.363·22-s + 1.67·23-s + 1.13·24-s + 0.333·25-s − 0.944·26-s + 1.74·27-s − 0.517·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8041 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8041 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.453201856\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.453201856\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| 43 | \( 1 + T \) |
good | 2 | \( 1 + 1.70T + 2T^{2} \) |
| 3 | \( 1 - 3.00T + 3T^{2} \) |
| 5 | \( 1 - 2.58T + 5T^{2} \) |
| 7 | \( 1 + 2.99T + 7T^{2} \) |
| 13 | \( 1 - 2.82T + 13T^{2} \) |
| 19 | \( 1 + 4.52T + 19T^{2} \) |
| 23 | \( 1 - 8.03T + 23T^{2} \) |
| 29 | \( 1 + 1.93T + 29T^{2} \) |
| 31 | \( 1 + 4.01T + 31T^{2} \) |
| 37 | \( 1 - 3.59T + 37T^{2} \) |
| 41 | \( 1 + 3.41T + 41T^{2} \) |
| 47 | \( 1 - 5.66T + 47T^{2} \) |
| 53 | \( 1 + 3.29T + 53T^{2} \) |
| 59 | \( 1 - 3.03T + 59T^{2} \) |
| 61 | \( 1 - 7.35T + 61T^{2} \) |
| 67 | \( 1 - 12.8T + 67T^{2} \) |
| 71 | \( 1 + 1.74T + 71T^{2} \) |
| 73 | \( 1 - 9.72T + 73T^{2} \) |
| 79 | \( 1 - 10.1T + 79T^{2} \) |
| 83 | \( 1 + 0.743T + 83T^{2} \) |
| 89 | \( 1 - 2.14T + 89T^{2} \) |
| 97 | \( 1 - 7.44T + 97T^{2} \) |
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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
−8.187789202328196787783461741205, −7.25798063038710659687248962631, −6.76470061855606270533246678043, −6.10878646149303074984238734919, −4.95930052179510013283776181413, −3.93039637762694812921034765437, −3.33183098026605657750096708688, −2.36636993222826690399338616588, −1.88224319253502857094130780630, −0.876519867817626614783528150571,
0.876519867817626614783528150571, 1.88224319253502857094130780630, 2.36636993222826690399338616588, 3.33183098026605657750096708688, 3.93039637762694812921034765437, 4.95930052179510013283776181413, 6.10878646149303074984238734919, 6.76470061855606270533246678043, 7.25798063038710659687248962631, 8.187789202328196787783461741205