| L(s) = 1 | − 2-s + 4-s − 8-s − 4·11-s − 13-s + 16-s − 4·17-s − 7·19-s + 4·22-s − 4·23-s + 26-s − 5·29-s − 4·31-s − 32-s + 4·34-s + 9·37-s + 7·38-s − 5·41-s − 10·43-s − 4·44-s + 4·46-s + 3·47-s − 52-s − 9·53-s + 5·58-s − 6·59-s − 4·61-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.353·8-s − 1.20·11-s − 0.277·13-s + 1/4·16-s − 0.970·17-s − 1.60·19-s + 0.852·22-s − 0.834·23-s + 0.196·26-s − 0.928·29-s − 0.718·31-s − 0.176·32-s + 0.685·34-s + 1.47·37-s + 1.13·38-s − 0.780·41-s − 1.52·43-s − 0.603·44-s + 0.589·46-s + 0.437·47-s − 0.138·52-s − 1.23·53-s + 0.656·58-s − 0.781·59-s − 0.512·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 286650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 286650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 11 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 7 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 9 T + p T^{2} \) |
| 41 | \( 1 + 5 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 + 7 T + p T^{2} \) |
| 71 | \( 1 - 15 T + p T^{2} \) |
| 73 | \( 1 + 12 T + p T^{2} \) |
| 79 | \( 1 - 7 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 - 16 T + p T^{2} \) |
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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
−13.05799364973366, −12.42466174452207, −12.07932779367833, −11.38806440357713, −11.02630051343118, −10.59863459980355, −10.33373704009236, −9.643601611529706, −9.336125922571513, −8.769490371760452, −8.304770273598535, −7.823837274482510, −7.619258561992710, −6.804778287824928, −6.509375844951822, −5.947046208690381, −5.484173368661286, −4.707486849864898, −4.491550292184878, −3.687788404775816, −3.148701145885126, −2.416088897063937, −2.057683569041358, −1.610370221511757, −0.4503535350100481, 0,
0.4503535350100481, 1.610370221511757, 2.057683569041358, 2.416088897063937, 3.148701145885126, 3.687788404775816, 4.491550292184878, 4.707486849864898, 5.484173368661286, 5.947046208690381, 6.509375844951822, 6.804778287824928, 7.619258561992710, 7.823837274482510, 8.304770273598535, 8.769490371760452, 9.336125922571513, 9.643601611529706, 10.33373704009236, 10.59863459980355, 11.02630051343118, 11.38806440357713, 12.07932779367833, 12.42466174452207, 13.05799364973366
