| L(s) = 1 | − 3-s − 2·4-s + 6·7-s + 2·12-s + 5·13-s − 6·17-s − 3·19-s − 6·21-s + 6·23-s + 27-s − 12·28-s + 6·29-s − 5·39-s − 18·41-s − 4·43-s + 17·49-s + 6·51-s − 10·52-s + 24·53-s + 3·57-s + 18·59-s − 61-s + 8·64-s − 6·67-s + 12·68-s − 6·69-s + 6·76-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 4-s + 2.26·7-s + 0.577·12-s + 1.38·13-s − 1.45·17-s − 0.688·19-s − 1.30·21-s + 1.25·23-s + 0.192·27-s − 2.26·28-s + 1.11·29-s − 0.800·39-s − 2.81·41-s − 0.609·43-s + 17/7·49-s + 0.840·51-s − 1.38·52-s + 3.29·53-s + 0.397·57-s + 2.34·59-s − 0.128·61-s + 64-s − 0.733·67-s + 1.45·68-s − 0.722·69-s + 0.688·76-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 950625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.618201877\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.618201877\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_2$ | \( 1 + T + T^{2} \) |
| 5 | | \( 1 \) |
| 13 | $C_2$ | \( 1 - 5 T + p T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 6 T + 19 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 3 T + 22 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 6 T + 13 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 6 T + 7 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 35 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 18 T + 149 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 4 T - 27 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 18 T + 167 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 71 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 18 T + 197 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 27 T + 340 T^{2} - 27 p T^{3} + p^{2} T^{4} \) |
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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
−10.29273330238597684440856561400, −10.09683535144635687611820333877, −8.970581611993185185369768182012, −8.810397854892319574645981941990, −8.620876657791883854407544653642, −8.424062741516042919181367775091, −7.900688629047452114625538474152, −7.12129245639779337268539584476, −6.89473734970337166177236000032, −6.40255826047643545743538754590, −5.80116215371448371432923662211, −5.16543727756025293640066872188, −4.97524432430367373473595144165, −4.70129807143628140411995943723, −3.97920720334513990865078097038, −3.86283752067353597273966652860, −2.76547040015412949209490289863, −2.03804722544250792827156278307, −1.45213911259607608662791721562, −0.67518652001974141877248812394,
0.67518652001974141877248812394, 1.45213911259607608662791721562, 2.03804722544250792827156278307, 2.76547040015412949209490289863, 3.86283752067353597273966652860, 3.97920720334513990865078097038, 4.70129807143628140411995943723, 4.97524432430367373473595144165, 5.16543727756025293640066872188, 5.80116215371448371432923662211, 6.40255826047643545743538754590, 6.89473734970337166177236000032, 7.12129245639779337268539584476, 7.900688629047452114625538474152, 8.424062741516042919181367775091, 8.620876657791883854407544653642, 8.810397854892319574645981941990, 8.970581611993185185369768182012, 10.09683535144635687611820333877, 10.29273330238597684440856561400
