| L(s) = 1 | + 2-s − 4-s − 3·8-s − 5·11-s + 4·13-s − 16-s + 4·17-s − 19-s − 5·22-s + 9·23-s + 4·26-s − 7·29-s + 3·31-s + 5·32-s + 4·34-s − 10·37-s − 38-s + 2·41-s + 4·43-s + 5·44-s + 9·46-s − 8·47-s − 7·49-s − 4·52-s − 11·53-s − 7·58-s − 8·59-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1/2·4-s − 1.06·8-s − 1.50·11-s + 1.10·13-s − 1/4·16-s + 0.970·17-s − 0.229·19-s − 1.06·22-s + 1.87·23-s + 0.784·26-s − 1.29·29-s + 0.538·31-s + 0.883·32-s + 0.685·34-s − 1.64·37-s − 0.162·38-s + 0.312·41-s + 0.609·43-s + 0.753·44-s + 1.32·46-s − 1.16·47-s − 49-s − 0.554·52-s − 1.51·53-s − 0.919·58-s − 1.04·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4275 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 2 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 9 T + p T^{2} \) |
| 29 | \( 1 + 7 T + p T^{2} \) |
| 31 | \( 1 - 3 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 11 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 - 13 T + p T^{2} \) |
| 67 | \( 1 - 9 T + p T^{2} \) |
| 71 | \( 1 + 10 T + p T^{2} \) |
| 73 | \( 1 + 5 T + p T^{2} \) |
| 79 | \( 1 + 15 T + p T^{2} \) |
| 83 | \( 1 + 9 T + p T^{2} \) |
| 89 | \( 1 + 3 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−8.136629484485467869473990139833, −7.27704990311364059872777446640, −6.38104958222058982677642160105, −5.45606563480218905168280098124, −5.22630651436391468557931895266, −4.28331882001411365877498297077, −3.32576523359967174858691921142, −2.89034509977461174442682170000, −1.39849965879244789298777254945, 0,
1.39849965879244789298777254945, 2.89034509977461174442682170000, 3.32576523359967174858691921142, 4.28331882001411365877498297077, 5.22630651436391468557931895266, 5.45606563480218905168280098124, 6.38104958222058982677642160105, 7.27704990311364059872777446640, 8.136629484485467869473990139833
