| L(s) = 1 | − 2-s + 6·3-s + 8·4-s − 5·5-s − 6·6-s − 23·8-s + 27·9-s + 5·10-s + 44·11-s + 48·12-s − 12·13-s − 30·15-s + 23·16-s − 24·17-s − 27·18-s − 114·19-s − 40·20-s − 44·22-s + 52·23-s − 138·24-s + 12·26-s + 270·27-s + 292·29-s + 30·30-s − 276·31-s − 184·32-s + 264·33-s + ⋯ |
| L(s) = 1 | − 0.353·2-s + 1.15·3-s + 4-s − 0.447·5-s − 0.408·6-s − 1.01·8-s + 9-s + 0.158·10-s + 1.20·11-s + 1.15·12-s − 0.256·13-s − 0.516·15-s + 0.359·16-s − 0.342·17-s − 0.353·18-s − 1.37·19-s − 0.447·20-s − 0.426·22-s + 0.471·23-s − 1.17·24-s + 0.0905·26-s + 1.92·27-s + 1.86·29-s + 0.182·30-s − 1.59·31-s − 1.01·32-s + 1.39·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 60025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60025 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.464732541\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.464732541\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | $C_2$ | \( 1 + p T + p^{2} T^{2} \) |
| 7 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 + T - 7 T^{2} + p^{3} T^{3} + p^{6} T^{4} \) |
| 3 | $C_2^2$ | \( 1 - 2 p T + p^{2} T^{2} - 2 p^{4} T^{3} + p^{6} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 4 p T + 5 p^{2} T^{2} - 4 p^{4} T^{3} + p^{6} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 6 T + p^{3} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 24 T - 4337 T^{2} + 24 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 6 p T + 17 p^{2} T^{2} + 6 p^{4} T^{3} + p^{6} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 52 T - 9463 T^{2} - 52 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 146 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 276 T + 46385 T^{2} + 276 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 210 T - 6553 T^{2} - 210 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 444 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 492 T + p^{3} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 612 T + 270721 T^{2} + 612 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 50 T - 146377 T^{2} + 50 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 294 T - 118943 T^{2} - 294 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 450 T - 24481 T^{2} - 450 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 668 T + 145461 T^{2} - 668 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 308 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 12 T - 388873 T^{2} - 12 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 596 T - 137823 T^{2} + 596 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 966 T + p^{3} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 408 T - 538505 T^{2} + 408 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 1200 T + p^{3} T^{2} )^{2} \) |
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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
−12.11320494608238821062645170583, −11.26412412306127042936721693954, −11.06512866181928825679626300375, −10.34524650508964855068021649734, −10.01085129781985605106409597646, −9.211876114583827543193620408850, −8.991447915261287580916313661425, −8.445240647554597092433178231772, −8.228281194079341844878086959110, −7.25942918957061387437352253589, −7.07741625644706790510009506654, −6.36434756850095356465865335019, −6.27175056855390404213661042118, −4.98767260432469650542098494388, −4.42334246317693178950190840185, −3.65618936143282872479750782116, −3.15199233694729586691514543406, −2.37571334434309979009210409639, −1.85246122579775927068741720375, −0.72906395855231197601581451403,
0.72906395855231197601581451403, 1.85246122579775927068741720375, 2.37571334434309979009210409639, 3.15199233694729586691514543406, 3.65618936143282872479750782116, 4.42334246317693178950190840185, 4.98767260432469650542098494388, 6.27175056855390404213661042118, 6.36434756850095356465865335019, 7.07741625644706790510009506654, 7.25942918957061387437352253589, 8.228281194079341844878086959110, 8.445240647554597092433178231772, 8.991447915261287580916313661425, 9.211876114583827543193620408850, 10.01085129781985605106409597646, 10.34524650508964855068021649734, 11.06512866181928825679626300375, 11.26412412306127042936721693954, 12.11320494608238821062645170583
