L(s) = 1 | − 2-s + 4-s − 7-s − 8-s + 4·11-s − 4·13-s + 14-s + 16-s − 8·17-s − 2·19-s − 4·22-s + 23-s + 4·26-s − 28-s − 2·29-s − 6·31-s − 32-s + 8·34-s + 10·37-s + 2·38-s − 6·41-s + 8·43-s + 4·44-s − 46-s + 6·47-s + 49-s − 4·52-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.377·7-s − 0.353·8-s + 1.20·11-s − 1.10·13-s + 0.267·14-s + 1/4·16-s − 1.94·17-s − 0.458·19-s − 0.852·22-s + 0.208·23-s + 0.784·26-s − 0.188·28-s − 0.371·29-s − 1.07·31-s − 0.176·32-s + 1.37·34-s + 1.64·37-s + 0.324·38-s − 0.937·41-s + 1.21·43-s + 0.603·44-s − 0.147·46-s + 0.875·47-s + 1/7·49-s − 0.554·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 72450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 23 | \( 1 - T \) |
good | 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 8 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 2 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 + 12 T + p T^{2} \) |
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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
−14.54235601674582, −13.85627005532077, −13.30619472788067, −12.72057465288680, −12.40819144982117, −11.72251602009976, −11.18938482692756, −10.99411447704185, −10.20742906674937, −9.708866385417657, −9.209862293243090, −8.918521628026152, −8.404700498170657, −7.584939472981258, −7.138818866316197, −6.721583481339550, −6.214407161066574, −5.620042377624317, −4.808640769641567, −4.177002167302508, −3.801164457178515, −2.777727412500854, −2.317062534028502, −1.718530442877149, −0.7575889121469554, 0,
0.7575889121469554, 1.718530442877149, 2.317062534028502, 2.777727412500854, 3.801164457178515, 4.177002167302508, 4.808640769641567, 5.620042377624317, 6.214407161066574, 6.721583481339550, 7.138818866316197, 7.584939472981258, 8.404700498170657, 8.918521628026152, 9.209862293243090, 9.708866385417657, 10.20742906674937, 10.99411447704185, 11.18938482692756, 11.72251602009976, 12.40819144982117, 12.72057465288680, 13.30619472788067, 13.85627005532077, 14.54235601674582