| L(s) = 1 | − 16·4-s − 1.44e3·11-s + 256·16-s − 712·19-s + 1.42e3·29-s + 1.69e3·31-s − 1.87e4·41-s + 2.30e4·44-s + 6.71e3·49-s + 1.58e4·59-s − 2.69e4·61-s − 4.09e3·64-s − 6.91e4·71-s + 1.13e4·76-s + 2.17e5·79-s + 2.16e4·89-s + 2.05e5·101-s + 4.07e5·109-s − 2.28e4·116-s + 1.23e6·121-s − 2.71e4·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | − 1/2·4-s − 3.58·11-s + 1/4·16-s − 0.452·19-s + 0.315·29-s + 0.316·31-s − 1.73·41-s + 1.79·44-s + 0.399·49-s + 0.592·59-s − 0.925·61-s − 1/8·64-s − 1.62·71-s + 0.226·76-s + 3.92·79-s + 0.289·89-s + 2.00·101-s + 3.28·109-s − 0.157·116-s + 7.65·121-s − 0.158·124-s + 5.50e−6·127-s + 5.09e−6·131-s + 4.55e−6·137-s + 4.38e−6·139-s + 3.69e−6·149-s + 3.56e−6·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 202500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 202500 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.5208708588\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5208708588\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_2$ | \( 1 + p^{4} T^{2} \) |
| 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( 1 - 6718 T^{2} + p^{10} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 720 T + p^{5} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 255382 T^{2} + p^{10} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 2115362 T^{2} + p^{10} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 356 T + p^{5} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 9632686 T^{2} + p^{10} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 714 T + p^{5} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 848 T + p^{5} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 10952710 T^{2} + p^{10} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 9354 T + p^{5} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 258542950 T^{2} + p^{10} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 334144414 T^{2} + p^{10} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 637411750 T^{2} + p^{10} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 7920 T + p^{5} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 13450 T + p^{5} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 1586856362 T^{2} + p^{10} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 34560 T + p^{5} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 3294299378 T^{2} + p^{10} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 108832 T + p^{5} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 7764275062 T^{2} + p^{10} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 10818 T + p^{5} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 17155161790 T^{2} + p^{10} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.33290483959618925906429145122, −10.26770853189966972950934696084, −9.823500313916693815500428692881, −9.055151433105476794080787779439, −8.635958159325272761808753046634, −8.201744231561279891433205443812, −7.76375865831578623937861604217, −7.56826367214387456129766100147, −6.88787508199301432168496655975, −6.20815496113314176008647297075, −5.65466827031898207981399327944, −5.24895572650162206218105796699, −4.76370193369470280747310058429, −4.53268944922333405944654841894, −3.32850252450212350343472278220, −3.21651082290675205624657864317, −2.25337921213869344216471863374, −2.13001826440033263115735507609, −0.841143819473082611230190392915, −0.21667118371993787629745191723,
0.21667118371993787629745191723, 0.841143819473082611230190392915, 2.13001826440033263115735507609, 2.25337921213869344216471863374, 3.21651082290675205624657864317, 3.32850252450212350343472278220, 4.53268944922333405944654841894, 4.76370193369470280747310058429, 5.24895572650162206218105796699, 5.65466827031898207981399327944, 6.20815496113314176008647297075, 6.88787508199301432168496655975, 7.56826367214387456129766100147, 7.76375865831578623937861604217, 8.201744231561279891433205443812, 8.635958159325272761808753046634, 9.055151433105476794080787779439, 9.823500313916693815500428692881, 10.26770853189966972950934696084, 10.33290483959618925906429145122
