| L(s) = 1 | − 2-s + 4-s + 5·7-s − 3·8-s + 11-s + 2·13-s − 5·14-s + 16-s + 17-s − 2·19-s − 22-s − 9·23-s − 2·26-s + 5·28-s − 6·29-s + 12·31-s + 32-s − 34-s + 9·37-s + 2·38-s + 3·41-s + 2·43-s + 44-s + 9·46-s − 14·47-s + 9·49-s + 2·52-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 1.88·7-s − 1.06·8-s + 0.301·11-s + 0.554·13-s − 1.33·14-s + 1/4·16-s + 0.242·17-s − 0.458·19-s − 0.213·22-s − 1.87·23-s − 0.392·26-s + 0.944·28-s − 1.11·29-s + 2.15·31-s + 0.176·32-s − 0.171·34-s + 1.47·37-s + 0.324·38-s + 0.468·41-s + 0.304·43-s + 0.150·44-s + 1.32·46-s − 2.04·47-s + 9/7·49-s + 0.277·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8555625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8555625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.283674034\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.283674034\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| 13 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 2 | $D_{4}$ | \( 1 + T + p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 5 T + 16 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - T + 18 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - T + 30 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 2 T + 22 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 9 T + 62 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 6 T + 50 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 37 | $D_{4}$ | \( 1 - 9 T + 56 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 3 T + 80 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 2 T + 70 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 14 T + 126 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 3 T + 70 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 61 | $D_{4}$ | \( 1 + T + 84 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 2 T - 18 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 25 T + 294 T^{2} - 25 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 79 | $D_{4}$ | \( 1 + 3 T + 122 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 106 T^{2} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 9 T + 160 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 5 T - 8 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
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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
−8.669802789411637988215261537362, −8.533768314648039439477523118229, −8.102709722292768512409734764168, −8.099892039610362740320999595846, −7.65698270380819280294888523190, −7.17780812217765016168627638595, −6.58334641197981666927230891619, −6.21920489423314869167322112560, −6.16745293178126316377869046072, −5.53968201568936277872707289203, −4.99614723879208031226798095871, −4.77725188200614312432605684970, −4.19376000668924508543344728870, −3.82557996177240797007616237295, −3.39478808647222406723076638630, −2.55024213533629244986545216833, −2.20456632767996594601131477438, −1.87759195166261424484489187406, −1.12380167435003520088277888790, −0.60372834305678554809227235106,
0.60372834305678554809227235106, 1.12380167435003520088277888790, 1.87759195166261424484489187406, 2.20456632767996594601131477438, 2.55024213533629244986545216833, 3.39478808647222406723076638630, 3.82557996177240797007616237295, 4.19376000668924508543344728870, 4.77725188200614312432605684970, 4.99614723879208031226798095871, 5.53968201568936277872707289203, 6.16745293178126316377869046072, 6.21920489423314869167322112560, 6.58334641197981666927230891619, 7.17780812217765016168627638595, 7.65698270380819280294888523190, 8.099892039610362740320999595846, 8.102709722292768512409734764168, 8.533768314648039439477523118229, 8.669802789411637988215261537362
