L(s) = 1 | − 2-s + 4-s − 8-s − 3·9-s − 2·13-s + 16-s + 6·17-s + 3·18-s + 4·19-s − 8·23-s + 2·26-s + 2·29-s − 31-s − 32-s − 6·34-s − 3·36-s − 10·37-s − 4·38-s − 6·41-s − 8·43-s + 8·46-s + 8·47-s − 7·49-s − 2·52-s + 6·53-s − 2·58-s − 12·59-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.353·8-s − 9-s − 0.554·13-s + 1/4·16-s + 1.45·17-s + 0.707·18-s + 0.917·19-s − 1.66·23-s + 0.392·26-s + 0.371·29-s − 0.179·31-s − 0.176·32-s − 1.02·34-s − 1/2·36-s − 1.64·37-s − 0.648·38-s − 0.937·41-s − 1.21·43-s + 1.17·46-s + 1.16·47-s − 49-s − 0.277·52-s + 0.824·53-s − 0.262·58-s − 1.56·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 \) |
| 31 | \( 1 + T \) |
good | 3 | \( 1 + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−9.024758063663975883170788252068, −8.211267180748869747116136469620, −7.68093503876904983414366404311, −6.74316601143042395037310474938, −5.75337536396212097143601099281, −5.15554787432694830495821906367, −3.64857119618832936893402436683, −2.82631239396687231926885349648, −1.57727173319487224423517628693, 0,
1.57727173319487224423517628693, 2.82631239396687231926885349648, 3.64857119618832936893402436683, 5.15554787432694830495821906367, 5.75337536396212097143601099281, 6.74316601143042395037310474938, 7.68093503876904983414366404311, 8.211267180748869747116136469620, 9.024758063663975883170788252068