| L(s) = 1 | − 2·3-s + 12·7-s − 5·9-s + 20·13-s − 4·19-s − 24·21-s + 18·25-s + 28·27-s + 44·31-s − 12·37-s − 40·39-s − 164·43-s + 10·49-s + 8·57-s − 172·61-s − 60·63-s − 4·67-s + 164·73-s − 36·75-s − 20·79-s − 11·81-s + 240·91-s − 88·93-s − 188·97-s + 268·103-s + 20·109-s + 24·111-s + ⋯ |
| L(s) = 1 | − 2/3·3-s + 12/7·7-s − 5/9·9-s + 1.53·13-s − 0.210·19-s − 8/7·21-s + 0.719·25-s + 1.03·27-s + 1.41·31-s − 0.324·37-s − 1.02·39-s − 3.81·43-s + 0.204·49-s + 8/57·57-s − 2.81·61-s − 0.952·63-s − 0.0597·67-s + 2.24·73-s − 0.479·75-s − 0.253·79-s − 0.135·81-s + 2.63·91-s − 0.946·93-s − 1.93·97-s + 2.60·103-s + 0.183·109-s + 8/37·111-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.129357969\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.129357969\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| 3 | $C_2$ | \( 1 + 2 T + p^{2} T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - 18 T^{2} + p^{4} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 6 T + p^{2} T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 210 T^{2} + p^{4} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 10 T + p^{2} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 66 T^{2} + p^{4} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 2 T + p^{2} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 930 T^{2} + p^{4} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 1394 T^{2} + p^{4} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 22 T + p^{2} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 6 T + p^{2} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 2210 T^{2} + p^{4} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 82 T + p^{2} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 190 T^{2} + p^{4} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 1746 T^{2} + p^{4} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 1554 T^{2} + p^{4} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 86 T + p^{2} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 2 T + p^{2} T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 + 5406 T^{2} + p^{4} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 82 T + p^{2} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 10 T + p^{2} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 8370 T^{2} + p^{4} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 14690 T^{2} + p^{4} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 94 T + p^{2} T^{2} )^{2} \) |
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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
−15.57043223853227462553880449452, −15.18205616010965058745776826752, −14.63570113284640684849674349504, −13.80983076217042636039033038115, −13.74779223154144646161891469990, −12.79875184868878539502629179602, −11.97297937000266713434486219270, −11.59147277940091212263005169755, −11.06628828728236644478383214993, −10.69952286346587240119431300380, −9.903901932416096870875403707062, −8.733397303057901341913019031213, −8.409602321260687544304099349977, −7.898390187659691057364673114554, −6.66243153714466825535656970955, −6.15665602481661924731916389292, −5.10001899190853530745711742341, −4.69649591590057934138421818576, −3.29890576054193911342403457106, −1.51516477393485154652426285439,
1.51516477393485154652426285439, 3.29890576054193911342403457106, 4.69649591590057934138421818576, 5.10001899190853530745711742341, 6.15665602481661924731916389292, 6.66243153714466825535656970955, 7.898390187659691057364673114554, 8.409602321260687544304099349977, 8.733397303057901341913019031213, 9.903901932416096870875403707062, 10.69952286346587240119431300380, 11.06628828728236644478383214993, 11.59147277940091212263005169755, 11.97297937000266713434486219270, 12.79875184868878539502629179602, 13.74779223154144646161891469990, 13.80983076217042636039033038115, 14.63570113284640684849674349504, 15.18205616010965058745776826752, 15.57043223853227462553880449452
