| L(s) = 1 | − 2·2-s + 3·4-s − 4·5-s + 2·7-s − 4·8-s − 3·9-s + 8·10-s + 4·11-s − 4·14-s + 5·16-s + 6·18-s − 12·20-s − 8·22-s + 10·23-s + 5·25-s + 6·28-s + 4·29-s − 8·31-s − 6·32-s − 8·35-s − 9·36-s + 12·37-s + 16·40-s − 12·41-s + 8·43-s + 12·44-s + 12·45-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 3/2·4-s − 1.78·5-s + 0.755·7-s − 1.41·8-s − 9-s + 2.52·10-s + 1.20·11-s − 1.06·14-s + 5/4·16-s + 1.41·18-s − 2.68·20-s − 1.70·22-s + 2.08·23-s + 25-s + 1.13·28-s + 0.742·29-s − 1.43·31-s − 1.06·32-s − 1.35·35-s − 3/2·36-s + 1.97·37-s + 2.52·40-s − 1.87·41-s + 1.21·43-s + 1.80·44-s + 1.78·45-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5597956 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5597956 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.026440177\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.026440177\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
| 13 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 4 T + 11 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 4 T + 14 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 22 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 10 T + 59 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 4 T + 14 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 8 T + 66 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 12 T + 98 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 12 T + 106 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $D_{4}$ | \( 1 - 4 T + 50 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 58 T^{2} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 16 T + 155 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 4 T + 99 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 4 T + 90 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 2 T + 131 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 8 T + 150 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 16 T + 174 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 58 T^{2} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 24 T + 310 T^{2} - 24 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 20 T + 246 T^{2} - 20 p T^{3} + p^{2} T^{4} \) |
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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
−8.985482468481825860122979382925, −8.821975471630970195759991037024, −8.501994808369764275341968301164, −8.014670133378411148586388291917, −7.65111527987991316272985377897, −7.35849111478084463727392316474, −7.31682761082116377290258510315, −6.49256881297276269871292154132, −6.25306406789705021879941006022, −5.92066937541887510734909664510, −5.02284102972145858711464536535, −4.90396048904913994711295690737, −4.33398205231086782078272777058, −3.73706899370529008336117419723, −3.27772453528571475243015271567, −3.11571271781561664056639438832, −2.21925446904970875547353981392, −1.77719600223170291185365882973, −0.73042288384820641168491640541, −0.71707200798988931466982101569,
0.71707200798988931466982101569, 0.73042288384820641168491640541, 1.77719600223170291185365882973, 2.21925446904970875547353981392, 3.11571271781561664056639438832, 3.27772453528571475243015271567, 3.73706899370529008336117419723, 4.33398205231086782078272777058, 4.90396048904913994711295690737, 5.02284102972145858711464536535, 5.92066937541887510734909664510, 6.25306406789705021879941006022, 6.49256881297276269871292154132, 7.31682761082116377290258510315, 7.35849111478084463727392316474, 7.65111527987991316272985377897, 8.014670133378411148586388291917, 8.501994808369764275341968301164, 8.821975471630970195759991037024, 8.985482468481825860122979382925
