| L(s) = 1 | + 2·5-s − 4·7-s − 3·9-s − 4·19-s + 8·23-s + 3·25-s − 4·29-s + 4·31-s − 8·35-s − 8·37-s − 10·41-s − 6·45-s − 4·47-s + 49-s − 8·53-s + 4·59-s − 10·61-s + 12·63-s + 8·67-s + 12·71-s − 16·73-s − 8·79-s + 8·83-s + 18·89-s − 8·95-s − 8·97-s − 2·101-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 1.51·7-s − 9-s − 0.917·19-s + 1.66·23-s + 3/5·25-s − 0.742·29-s + 0.718·31-s − 1.35·35-s − 1.31·37-s − 1.56·41-s − 0.894·45-s − 0.583·47-s + 1/7·49-s − 1.09·53-s + 0.520·59-s − 1.28·61-s + 1.51·63-s + 0.977·67-s + 1.42·71-s − 1.87·73-s − 0.900·79-s + 0.878·83-s + 1.90·89-s − 0.820·95-s − 0.812·97-s − 0.199·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23425600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23425600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 11 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 4 T + 15 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 23 | $D_{4}$ | \( 1 - 8 T + 50 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 4 T + 14 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 4 T + 54 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 10 T + 59 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 83 T^{2} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 4 T + 23 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 8 T + 110 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 4 T + 110 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 10 T + 135 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 8 T + 147 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 12 T + 166 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 16 T + 162 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 8 T + 66 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 8 T + 170 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 18 T + 211 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 8 T + 102 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
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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
−8.105135227094967112822494959093, −7.84753234068020269232166021703, −7.16841920838354754681495001969, −6.89612588365918559484952764328, −6.53709702176599717484256472948, −6.41788252781019014420718940682, −5.94947741527024085178450466362, −5.64039860097785044624081189762, −5.07257763953046251388323293577, −4.97973766081770868354880007044, −4.48393011747447579454550068455, −3.69005270860617318785825050758, −3.41388995407231539535170652027, −3.20209625287725742123447739894, −2.53460810868291701898419763630, −2.42808760330202337110595724028, −1.60786919522234835516611943545, −1.18838819117394696200069298759, 0, 0,
1.18838819117394696200069298759, 1.60786919522234835516611943545, 2.42808760330202337110595724028, 2.53460810868291701898419763630, 3.20209625287725742123447739894, 3.41388995407231539535170652027, 3.69005270860617318785825050758, 4.48393011747447579454550068455, 4.97973766081770868354880007044, 5.07257763953046251388323293577, 5.64039860097785044624081189762, 5.94947741527024085178450466362, 6.41788252781019014420718940682, 6.53709702176599717484256472948, 6.89612588365918559484952764328, 7.16841920838354754681495001969, 7.84753234068020269232166021703, 8.105135227094967112822494959093
