| L(s) = 1 | + 1.56i·3-s + 2.56i·7-s + 0.561·9-s + 2·11-s + 3.56i·13-s + 2.56i·17-s − 6·19-s − 4·21-s − i·23-s + 5.56i·27-s − 6.12·29-s + 7.24·31-s + 3.12i·33-s − 4.56i·37-s − 5.56·39-s + ⋯ |
| L(s) = 1 | + 0.901i·3-s + 0.968i·7-s + 0.187·9-s + 0.603·11-s + 0.987i·13-s + 0.621i·17-s − 1.37·19-s − 0.872·21-s − 0.208i·23-s + 1.07i·27-s − 1.13·29-s + 1.30·31-s + 0.543i·33-s − 0.749i·37-s − 0.890·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.519273375\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.519273375\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 23 | \( 1 + iT \) |
| good | 3 | \( 1 - 1.56iT - 3T^{2} \) |
| 7 | \( 1 - 2.56iT - 7T^{2} \) |
| 11 | \( 1 - 2T + 11T^{2} \) |
| 13 | \( 1 - 3.56iT - 13T^{2} \) |
| 17 | \( 1 - 2.56iT - 17T^{2} \) |
| 19 | \( 1 + 6T + 19T^{2} \) |
| 29 | \( 1 + 6.12T + 29T^{2} \) |
| 31 | \( 1 - 7.24T + 31T^{2} \) |
| 37 | \( 1 + 4.56iT - 37T^{2} \) |
| 41 | \( 1 - 4.12T + 41T^{2} \) |
| 43 | \( 1 - 43T^{2} \) |
| 47 | \( 1 - 4.68iT - 47T^{2} \) |
| 53 | \( 1 - 4.56iT - 53T^{2} \) |
| 59 | \( 1 - 3.68T + 59T^{2} \) |
| 61 | \( 1 + 7.12T + 61T^{2} \) |
| 67 | \( 1 + 8.56iT - 67T^{2} \) |
| 71 | \( 1 - 10.1T + 71T^{2} \) |
| 73 | \( 1 - 4.43iT - 73T^{2} \) |
| 79 | \( 1 + 4.87T + 79T^{2} \) |
| 83 | \( 1 - 13.9iT - 83T^{2} \) |
| 89 | \( 1 + 14.2T + 89T^{2} \) |
| 97 | \( 1 + 13.1iT - 97T^{2} \) |
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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
−9.259432319732943686558634576995, −8.843496249166157591224255948331, −7.983822562655058200323374638573, −6.83837185917807805546585086895, −6.20157648963508884697311454059, −5.34506170508296469213931086513, −4.26527282186323017147359551419, −4.00428535033584133867341723346, −2.62845434297040578089079416770, −1.64604722216838827234324406590,
0.52818152986298899885087905296, 1.51112087306479009439007422655, 2.65485863579735171951860026419, 3.83383714250876091929595743266, 4.54208524705766411609994900361, 5.69174898879537983679204803186, 6.59907380602805694765537131085, 7.06745213933410903875481448899, 7.84735525707203661722662645031, 8.425794017452273794151957018317
