| L(s) = 1 | − 2·5-s − 2·7-s + 2·9-s + 2·11-s + 4·13-s + 4·17-s + 8·23-s + 3·25-s − 4·29-s − 8·31-s + 4·35-s + 12·37-s + 12·41-s + 8·43-s − 4·45-s + 3·49-s − 4·53-s − 4·55-s + 8·59-s + 12·61-s − 4·63-s − 8·65-s − 12·73-s − 4·77-s − 5·81-s − 8·85-s − 12·89-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 0.755·7-s + 2/3·9-s + 0.603·11-s + 1.10·13-s + 0.970·17-s + 1.66·23-s + 3/5·25-s − 0.742·29-s − 1.43·31-s + 0.676·35-s + 1.97·37-s + 1.87·41-s + 1.21·43-s − 0.596·45-s + 3/7·49-s − 0.549·53-s − 0.539·55-s + 1.04·59-s + 1.53·61-s − 0.503·63-s − 0.992·65-s − 1.40·73-s − 0.455·77-s − 5/9·81-s − 0.867·85-s − 1.27·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 37945600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37945600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.294733428\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.294733428\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 7 | $C_1$ | \( ( 1 + T )^{2} \) |
| 11 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 4 T + 22 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 4 T + 30 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 31 | $D_{4}$ | \( 1 + 8 T + 70 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 12 T + 78 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 12 T + 110 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 8 T + 70 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 86 T^{2} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 4 T + 78 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 8 T + 126 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 12 T + 150 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 102 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 110 T^{2} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 12 T + 174 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 134 T^{2} + p^{2} T^{4} \) |
| 89 | $C_4$ | \( 1 + 12 T + 86 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 4 T + 70 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
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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
−8.022813675981981209254701483117, −7.908123402405193364210459148466, −7.52910573478590516383473400161, −7.08235557852766573879249711587, −6.97192237442331320853536796159, −6.60369776744075058562709058375, −6.00709493975579705325297693642, −5.78861259835636369185549575833, −5.51248926678965835630395708089, −5.03669621774929930212642547482, −4.34020811622577044259112023816, −4.11643560545996901050032310893, −3.98819276744448794967835378609, −3.48187291089578236478131044922, −2.93069389027776955844902958256, −2.86534155054653473745408298368, −2.06020887469434310903254983011, −1.41072520765682018409141049351, −0.939476954827322869412651734936, −0.60185193535486940080022239391,
0.60185193535486940080022239391, 0.939476954827322869412651734936, 1.41072520765682018409141049351, 2.06020887469434310903254983011, 2.86534155054653473745408298368, 2.93069389027776955844902958256, 3.48187291089578236478131044922, 3.98819276744448794967835378609, 4.11643560545996901050032310893, 4.34020811622577044259112023816, 5.03669621774929930212642547482, 5.51248926678965835630395708089, 5.78861259835636369185549575833, 6.00709493975579705325297693642, 6.60369776744075058562709058375, 6.97192237442331320853536796159, 7.08235557852766573879249711587, 7.52910573478590516383473400161, 7.908123402405193364210459148466, 8.022813675981981209254701483117
