| L(s) = 1 | + 2·11-s − 6·13-s + 9·17-s − 3·19-s − 23-s − 2·29-s − 9·31-s − 2·37-s + 2·41-s − 8·43-s + 5·47-s − 14·49-s + 11·53-s + 2·59-s + 61-s − 12·67-s − 6·71-s − 11·79-s + 17·83-s − 18·89-s − 4·97-s − 24·101-s − 2·103-s + 107-s + 21·109-s + 113-s − 14·121-s + ⋯ |
| L(s) = 1 | + 0.603·11-s − 1.66·13-s + 2.18·17-s − 0.688·19-s − 0.208·23-s − 0.371·29-s − 1.61·31-s − 0.328·37-s + 0.312·41-s − 1.21·43-s + 0.729·47-s − 2·49-s + 1.51·53-s + 0.260·59-s + 0.128·61-s − 1.46·67-s − 0.712·71-s − 1.23·79-s + 1.86·83-s − 1.90·89-s − 0.406·97-s − 2.38·101-s − 0.197·103-s + 0.0966·107-s + 2.01·109-s + 0.0940·113-s − 1.27·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 81000000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81000000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $D_{4}$ | \( 1 - 2 T + 18 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 6 T + 30 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 9 T + 43 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 3 T + 39 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + T + 35 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 2 T + 54 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 9 T + 81 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 2 T + 70 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 2 T + 78 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 8 T + 82 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 5 T + 69 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 11 T + 135 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 2 T - 6 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - T + 121 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 12 T + 90 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_4$ | \( 1 + 6 T + 26 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 66 T^{2} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 11 T + 187 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 17 T + 177 T^{2} - 17 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 18 T + 254 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 4 T + 118 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
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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
−7.52717325159768530270969810696, −7.28803927068176901567528403245, −6.85840273628066243864076422096, −6.75747143394682731722574367243, −6.09828004558912712089416354179, −5.79271710041246260187830179398, −5.47023078486040518970966375597, −5.29408613760418294991650863625, −4.63128155981621757644337285054, −4.62843278652524055023556625283, −3.85858639540731446301337743784, −3.76529344533187780122124035172, −3.20411575180936709261441894958, −2.96959269291306325093127058862, −2.30346121974547957596504213333, −2.07410347801584169966326927247, −1.33166903669837875939490112405, −1.21521918029060498041211452602, 0, 0,
1.21521918029060498041211452602, 1.33166903669837875939490112405, 2.07410347801584169966326927247, 2.30346121974547957596504213333, 2.96959269291306325093127058862, 3.20411575180936709261441894958, 3.76529344533187780122124035172, 3.85858639540731446301337743784, 4.62843278652524055023556625283, 4.63128155981621757644337285054, 5.29408613760418294991650863625, 5.47023078486040518970966375597, 5.79271710041246260187830179398, 6.09828004558912712089416354179, 6.75747143394682731722574367243, 6.85840273628066243864076422096, 7.28803927068176901567528403245, 7.52717325159768530270969810696
