L(s) = 1 | + (5 + 10i)5-s − 26i·7-s − 28·11-s − 12i·13-s + 64i·17-s − 60·19-s + 58i·23-s + (−75 + 100i)25-s + 90·29-s + 128·31-s + (260 − 130i)35-s + 236i·37-s − 242·41-s + 362i·43-s + 226i·47-s + ⋯ |
L(s) = 1 | + (0.447 + 0.894i)5-s − 1.40i·7-s − 0.767·11-s − 0.256i·13-s + 0.913i·17-s − 0.724·19-s + 0.525i·23-s + (−0.599 + 0.800i)25-s + 0.576·29-s + 0.741·31-s + (1.25 − 0.627i)35-s + 1.04i·37-s − 0.921·41-s + 1.28i·43-s + 0.701i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.109124206\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.109124206\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-5 - 10i)T \) |
good | 7 | \( 1 + 26iT - 343T^{2} \) |
| 11 | \( 1 + 28T + 1.33e3T^{2} \) |
| 13 | \( 1 + 12iT - 2.19e3T^{2} \) |
| 17 | \( 1 - 64iT - 4.91e3T^{2} \) |
| 19 | \( 1 + 60T + 6.85e3T^{2} \) |
| 23 | \( 1 - 58iT - 1.21e4T^{2} \) |
| 29 | \( 1 - 90T + 2.43e4T^{2} \) |
| 31 | \( 1 - 128T + 2.97e4T^{2} \) |
| 37 | \( 1 - 236iT - 5.06e4T^{2} \) |
| 41 | \( 1 + 242T + 6.89e4T^{2} \) |
| 43 | \( 1 - 362iT - 7.95e4T^{2} \) |
| 47 | \( 1 - 226iT - 1.03e5T^{2} \) |
| 53 | \( 1 + 108iT - 1.48e5T^{2} \) |
| 59 | \( 1 - 20T + 2.05e5T^{2} \) |
| 61 | \( 1 - 542T + 2.26e5T^{2} \) |
| 67 | \( 1 - 434iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 1.12e3T + 3.57e5T^{2} \) |
| 73 | \( 1 + 632iT - 3.89e5T^{2} \) |
| 79 | \( 1 + 720T + 4.93e5T^{2} \) |
| 83 | \( 1 - 478iT - 5.71e5T^{2} \) |
| 89 | \( 1 + 490T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.45e3iT - 9.12e5T^{2} \) |
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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
−10.34581645352371701977329874842, −9.816286128502080798848836558359, −8.380848014452515487858073964046, −7.64506717155168582949716136865, −6.77120246454101472728929849329, −6.04820468977411626267255619053, −4.77629299257984150591043866522, −3.72567478845097658737359769558, −2.71373281496499454479438303591, −1.32112351694799658375442404034,
0.30037764341700008923397956321, 1.95500920297652367921591402519, 2.77194529621760488438117723485, 4.43761069914329256552356823254, 5.29149880757721249256393405776, 5.92820828614388235569216177166, 7.06836026275220779523099613789, 8.437400582584077383016130373626, 8.695598428592748740083714589366, 9.646052268135818892784560341564