| L(s) = 1 | − 2-s + 3·3-s − 2·4-s + 5-s − 3·6-s + 3·8-s + 2·9-s − 10-s − 6·12-s + 6·13-s + 3·15-s + 16-s + 6·17-s − 2·18-s + 3·19-s − 2·20-s + 3·23-s + 9·24-s − 8·25-s − 6·26-s − 6·27-s + 9·29-s − 3·30-s + 17·31-s − 2·32-s − 6·34-s − 4·36-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.73·3-s − 4-s + 0.447·5-s − 1.22·6-s + 1.06·8-s + 2/3·9-s − 0.316·10-s − 1.73·12-s + 1.66·13-s + 0.774·15-s + 1/4·16-s + 1.45·17-s − 0.471·18-s + 0.688·19-s − 0.447·20-s + 0.625·23-s + 1.83·24-s − 8/5·25-s − 1.17·26-s − 1.15·27-s + 1.67·29-s − 0.547·30-s + 3.05·31-s − 0.353·32-s − 1.02·34-s − 2/3·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4036081 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4036081 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.356581721\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.356581721\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 7 | | \( 1 \) |
| 41 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 2 | $D_{4}$ | \( 1 + T + 3 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 3 | $D_{4}$ | \( 1 - p T + 7 T^{2} - p^{2} T^{3} + p^{2} T^{4} \) |
| 5 | $D_{4}$ | \( 1 - T + 9 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 17 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 17 | $D_{4}$ | \( 1 - 6 T + 23 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 3 T + 29 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 3 T + 47 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 9 T + 67 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 17 T + 133 T^{2} - 17 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 2 T + 30 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 2 T + 7 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 2 T + 90 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 17 T + 147 T^{2} + 17 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 13 T + 159 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2} \) |
| 67 | $D_{4}$ | \( 1 - 11 T + 163 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 4 T + 141 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 15 T + p T^{2} )^{2} \) |
| 79 | $D_{4}$ | \( 1 + 2 T + 114 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 6 T + 95 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 9 T + 97 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 17 T + 255 T^{2} - 17 p T^{3} + p^{2} T^{4} \) |
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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
−9.241257455666600299579759896719, −8.979173908219833097796157929492, −8.373717155885574710489964394010, −8.315294105987459104145178539219, −8.028635028771748010798222393763, −7.932690013042355845625024294096, −7.26113324420639451125554182076, −6.44125215526054741261953371540, −6.34037825940922110337747113072, −5.88772494991849005091051593523, −5.16224165825303774663643470255, −4.96986321904285540696941499465, −4.33562412179366437576922074007, −3.82899753865493452672851720748, −3.30275508726194121211852468520, −3.20415124671576051441965740403, −2.54476667461716509288001296101, −1.97495888608341075472217053728, −1.05055021946268001399832260043, −0.879239708828087375650747272474,
0.879239708828087375650747272474, 1.05055021946268001399832260043, 1.97495888608341075472217053728, 2.54476667461716509288001296101, 3.20415124671576051441965740403, 3.30275508726194121211852468520, 3.82899753865493452672851720748, 4.33562412179366437576922074007, 4.96986321904285540696941499465, 5.16224165825303774663643470255, 5.88772494991849005091051593523, 6.34037825940922110337747113072, 6.44125215526054741261953371540, 7.26113324420639451125554182076, 7.932690013042355845625024294096, 8.028635028771748010798222393763, 8.315294105987459104145178539219, 8.373717155885574710489964394010, 8.979173908219833097796157929492, 9.241257455666600299579759896719
