| L(s) = 1 | − 2·4-s − 7-s + 3·11-s − 5·13-s + 4·16-s + 3·17-s + 2·19-s − 6·23-s + 2·28-s − 3·29-s − 4·31-s − 2·37-s + 12·41-s + 10·43-s − 6·44-s + 9·47-s + 49-s + 10·52-s + 12·53-s + 8·61-s − 8·64-s + 4·67-s − 6·68-s − 2·73-s − 4·76-s − 3·77-s − 79-s + ⋯ |
| L(s) = 1 | − 4-s − 0.377·7-s + 0.904·11-s − 1.38·13-s + 16-s + 0.727·17-s + 0.458·19-s − 1.25·23-s + 0.377·28-s − 0.557·29-s − 0.718·31-s − 0.328·37-s + 1.87·41-s + 1.52·43-s − 0.904·44-s + 1.31·47-s + 1/7·49-s + 1.38·52-s + 1.64·53-s + 1.02·61-s − 64-s + 0.488·67-s − 0.727·68-s − 0.234·73-s − 0.458·76-s − 0.341·77-s − 0.112·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.138679967\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.138679967\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 2 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 - T + p T^{2} \) |
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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
−9.467462389572036350103527465528, −8.846894389721446221328072634367, −7.73929750781885351517852687666, −7.24281290023688107365847358204, −5.95865885309513387304829184659, −5.36755822629614084385217196812, −4.25386150379293419498976865827, −3.67606993779998961598679741611, −2.35094220433354059590404276967, −0.75515587387361470267464831173,
0.75515587387361470267464831173, 2.35094220433354059590404276967, 3.67606993779998961598679741611, 4.25386150379293419498976865827, 5.36755822629614084385217196812, 5.95865885309513387304829184659, 7.24281290023688107365847358204, 7.73929750781885351517852687666, 8.846894389721446221328072634367, 9.467462389572036350103527465528
