L(s) = 1 | − 2·3-s − 4·5-s + 9-s + 6·11-s − 2·13-s + 8·15-s + 17-s − 4·23-s + 11·25-s + 4·27-s − 8·29-s − 12·33-s − 4·37-s + 4·39-s + 2·41-s + 8·43-s − 4·45-s + 8·47-s − 2·51-s + 6·53-s − 24·55-s − 4·59-s − 8·61-s + 8·65-s + 16·67-s + 8·69-s + 4·71-s + ⋯ |
L(s) = 1 | − 1.15·3-s − 1.78·5-s + 1/3·9-s + 1.80·11-s − 0.554·13-s + 2.06·15-s + 0.242·17-s − 0.834·23-s + 11/5·25-s + 0.769·27-s − 1.48·29-s − 2.08·33-s − 0.657·37-s + 0.640·39-s + 0.312·41-s + 1.21·43-s − 0.596·45-s + 1.16·47-s − 0.280·51-s + 0.824·53-s − 3.23·55-s − 0.520·59-s − 1.02·61-s + 0.992·65-s + 1.95·67-s + 0.963·69-s + 0.474·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 53312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 53312 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| 17 | \( 1 - T \) |
good | 3 | \( 1 + 2 T + p T^{2} \) |
| 5 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 16 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−14.68436812864008, −14.34897722117898, −13.84423667183349, −12.80692602211580, −12.38445911852601, −12.03242119334227, −11.72840113711620, −11.18360828105503, −10.94162651003039, −10.20861808651177, −9.519170918497881, −8.888140579561164, −8.537401119940145, −7.570274649315669, −7.447792686254153, −6.780290332934965, −6.223076236274977, −5.649964405445621, −4.997139522218325, −4.257232136203193, −3.968979593129054, −3.461195663046533, −2.506101775319304, −1.429973537547471, −0.7019466687430296, 0,
0.7019466687430296, 1.429973537547471, 2.506101775319304, 3.461195663046533, 3.968979593129054, 4.257232136203193, 4.997139522218325, 5.649964405445621, 6.223076236274977, 6.780290332934965, 7.447792686254153, 7.570274649315669, 8.537401119940145, 8.888140579561164, 9.519170918497881, 10.20861808651177, 10.94162651003039, 11.18360828105503, 11.72840113711620, 12.03242119334227, 12.38445911852601, 12.80692602211580, 13.84423667183349, 14.34897722117898, 14.68436812864008