| L(s) = 1 | − 3-s + 4-s − 5-s − 4·9-s + 2·11-s − 12-s − 2·13-s + 15-s − 3·16-s − 10·17-s − 19-s − 20-s − 8·25-s + 6·27-s − 11·29-s + 4·31-s − 2·33-s − 4·36-s + 11·37-s + 2·39-s − 41-s − 17·43-s + 2·44-s + 4·45-s + 3·48-s + 6·49-s + 10·51-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/2·4-s − 0.447·5-s − 4/3·9-s + 0.603·11-s − 0.288·12-s − 0.554·13-s + 0.258·15-s − 3/4·16-s − 2.42·17-s − 0.229·19-s − 0.223·20-s − 8/5·25-s + 1.15·27-s − 2.04·29-s + 0.718·31-s − 0.348·33-s − 2/3·36-s + 1.80·37-s + 0.320·39-s − 0.156·41-s − 2.59·43-s + 0.301·44-s + 0.596·45-s + 0.433·48-s + 6/7·49-s + 1.40·51-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1234321 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1234321 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 11 | $C_1$ | \( ( 1 - T )^{2} \) |
| 101 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 3 | $D_{4}$ | \( 1 + T + 5 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 5 | $D_{4}$ | \( 1 + T + 9 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 2 T + 22 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 10 T + 54 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + T + 37 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 11 T + 77 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 37 | $D_{4}$ | \( 1 - 11 T + 93 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + T + 81 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 17 T + 157 T^{2} + 17 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 74 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 86 T^{2} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 3 T + 119 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 9 T + 141 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 21 T + 243 T^{2} - 21 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 20 T + 222 T^{2} + 20 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 13 T + 177 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 7 T + 159 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 12 T + 122 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 158 T^{2} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 29 T + 393 T^{2} - 29 p T^{3} + p^{2} T^{4} \) |
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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
−9.496003027469591242034318238776, −9.225691612837091421422074490893, −8.821712679337880872905584202336, −8.527606367449655033775687693854, −7.82737792946201774908198920883, −7.66927293578643483834462079004, −7.07580783852639855296210614436, −6.42843436888685003410425968178, −6.39440170466707998259098911203, −6.05151173609962738041699847284, −5.21627209398535330942100604017, −5.06004733179533860752870549064, −4.19302100863895195653034604025, −4.13233144760320956833283891348, −3.36714862792359978865555089484, −2.54946760974473699959776258011, −2.32487111917228566213996935720, −1.62005765415946158634033403060, 0, 0,
1.62005765415946158634033403060, 2.32487111917228566213996935720, 2.54946760974473699959776258011, 3.36714862792359978865555089484, 4.13233144760320956833283891348, 4.19302100863895195653034604025, 5.06004733179533860752870549064, 5.21627209398535330942100604017, 6.05151173609962738041699847284, 6.39440170466707998259098911203, 6.42843436888685003410425968178, 7.07580783852639855296210614436, 7.66927293578643483834462079004, 7.82737792946201774908198920883, 8.527606367449655033775687693854, 8.821712679337880872905584202336, 9.225691612837091421422074490893, 9.496003027469591242034318238776
