L(s) = 1 | + 3-s − 7-s + 9-s − 5·11-s − 7·13-s − 4·17-s + 19-s − 21-s + 4·23-s + 27-s + 5·29-s − 10·31-s − 5·33-s + 10·37-s − 7·39-s + 43-s + 47-s − 6·49-s − 4·51-s + 12·53-s + 57-s − 4·59-s + 8·61-s − 63-s − 2·67-s + 4·69-s − 12·71-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 0.377·7-s + 1/3·9-s − 1.50·11-s − 1.94·13-s − 0.970·17-s + 0.229·19-s − 0.218·21-s + 0.834·23-s + 0.192·27-s + 0.928·29-s − 1.79·31-s − 0.870·33-s + 1.64·37-s − 1.12·39-s + 0.152·43-s + 0.145·47-s − 6/7·49-s − 0.560·51-s + 1.64·53-s + 0.132·57-s − 0.520·59-s + 1.02·61-s − 0.125·63-s − 0.244·67-s + 0.481·69-s − 1.42·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 206400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 206400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 43 | \( 1 - T \) |
good | 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 + 7 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 5 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 47 | \( 1 - T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + 7 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 7 T + p T^{2} \) |
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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
−13.14621762201962, −13.02239697763680, −12.39856946172592, −12.01506642755149, −11.33202155021749, −10.87085808882870, −10.34117313933837, −10.01998920142158, −9.406865017667641, −9.164527523777338, −8.541907115503993, −7.935854979458315, −7.570596049078719, −7.132742995110784, −6.758846613888885, −5.997302280406184, −5.307752986075433, −5.052100914708845, −4.453273822491883, −3.951734588833711, −3.067554037387362, −2.652564502902094, −2.426064631622715, −1.682197296630926, −0.6357819096685552, 0,
0.6357819096685552, 1.682197296630926, 2.426064631622715, 2.652564502902094, 3.067554037387362, 3.951734588833711, 4.453273822491883, 5.052100914708845, 5.307752986075433, 5.997302280406184, 6.758846613888885, 7.132742995110784, 7.570596049078719, 7.935854979458315, 8.541907115503993, 9.164527523777338, 9.406865017667641, 10.01998920142158, 10.34117313933837, 10.87085808882870, 11.33202155021749, 12.01506642755149, 12.39856946172592, 13.02239697763680, 13.14621762201962