| L(s) = 1 | − 2·3-s + 2·7-s + 9-s − 6·11-s + 13-s − 6·17-s − 2·19-s − 4·21-s + 4·27-s − 9·29-s + 4·31-s + 12·33-s − 2·37-s − 2·39-s − 9·41-s − 4·43-s + 6·47-s − 3·49-s + 12·51-s − 3·53-s + 4·57-s − 6·59-s − 61-s + 2·63-s + 8·67-s − 6·71-s − 11·73-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 0.755·7-s + 1/3·9-s − 1.80·11-s + 0.277·13-s − 1.45·17-s − 0.458·19-s − 0.872·21-s + 0.769·27-s − 1.67·29-s + 0.718·31-s + 2.08·33-s − 0.328·37-s − 0.320·39-s − 1.40·41-s − 0.609·43-s + 0.875·47-s − 3/7·49-s + 1.68·51-s − 0.412·53-s + 0.529·57-s − 0.781·59-s − 0.128·61-s + 0.251·63-s + 0.977·67-s − 0.712·71-s − 1.28·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 211600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 211600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 23 | \( 1 \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 11 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 9 T + p T^{2} \) |
| 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
−13.21872053520519, −12.75015252078167, −12.30161343316042, −11.70023596184909, −11.15267074221016, −11.12975319021260, −10.56429234422209, −10.19733119641559, −9.633264022283331, −8.840471228688927, −8.464385875393903, −8.119555391033177, −7.410689618719008, −7.050342266255644, −6.429624881276346, −5.913676068452563, −5.462768006127812, −5.045736670349548, −4.590252842621830, −4.166177658401134, −3.232290010024092, −2.717350621589218, −1.974662014549981, −1.587309546587202, −0.4782209592383485, 0,
0.4782209592383485, 1.587309546587202, 1.974662014549981, 2.717350621589218, 3.232290010024092, 4.166177658401134, 4.590252842621830, 5.045736670349548, 5.462768006127812, 5.913676068452563, 6.429624881276346, 7.050342266255644, 7.410689618719008, 8.119555391033177, 8.464385875393903, 8.840471228688927, 9.633264022283331, 10.19733119641559, 10.56429234422209, 11.12975319021260, 11.15267074221016, 11.70023596184909, 12.30161343316042, 12.75015252078167, 13.21872053520519
