| L(s) = 1 | + 5.37e3·5-s + 2.77e4·7-s + 6.37e5·11-s + 7.66e5·13-s − 3.08e6·17-s + 1.95e7·19-s + 1.53e7·23-s − 1.99e7·25-s − 1.07e7·29-s + 5.09e7·31-s + 1.49e8·35-s + 6.64e8·37-s − 8.98e8·41-s + 9.57e8·43-s − 1.55e9·47-s − 1.20e9·49-s − 3.79e9·53-s + 3.42e9·55-s + 5.55e8·59-s + 4.95e9·61-s + 4.11e9·65-s − 5.29e9·67-s − 1.48e10·71-s + 1.39e10·73-s + 1.77e10·77-s − 3.72e9·79-s + 8.76e9·83-s + ⋯ |
| L(s) = 1 | + 0.768·5-s + 0.624·7-s + 1.19·11-s + 0.572·13-s − 0.526·17-s + 1.80·19-s + 0.496·23-s − 0.409·25-s − 0.0973·29-s + 0.319·31-s + 0.479·35-s + 1.57·37-s − 1.21·41-s + 0.993·43-s − 0.989·47-s − 0.610·49-s − 1.24·53-s + 0.917·55-s + 0.101·59-s + 0.750·61-s + 0.439·65-s − 0.478·67-s − 0.975·71-s + 0.788·73-s + 0.745·77-s − 0.136·79-s + 0.244·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(3.528271390\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.528271390\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 1074 p T + p^{11} T^{2} \) |
| 7 | \( 1 - 27760 T + p^{11} T^{2} \) |
| 11 | \( 1 - 637836 T + p^{11} T^{2} \) |
| 13 | \( 1 - 766214 T + p^{11} T^{2} \) |
| 17 | \( 1 + 3084354 T + p^{11} T^{2} \) |
| 19 | \( 1 - 1026916 p T + p^{11} T^{2} \) |
| 23 | \( 1 - 15312360 T + p^{11} T^{2} \) |
| 29 | \( 1 + 10751262 T + p^{11} T^{2} \) |
| 31 | \( 1 - 50937400 T + p^{11} T^{2} \) |
| 37 | \( 1 - 664740830 T + p^{11} T^{2} \) |
| 41 | \( 1 + 898833450 T + p^{11} T^{2} \) |
| 43 | \( 1 - 957947188 T + p^{11} T^{2} \) |
| 47 | \( 1 + 1555741344 T + p^{11} T^{2} \) |
| 53 | \( 1 + 3792417030 T + p^{11} T^{2} \) |
| 59 | \( 1 - 555306924 T + p^{11} T^{2} \) |
| 61 | \( 1 - 4950420998 T + p^{11} T^{2} \) |
| 67 | \( 1 + 5292399284 T + p^{11} T^{2} \) |
| 71 | \( 1 + 14831086248 T + p^{11} T^{2} \) |
| 73 | \( 1 - 13971005210 T + p^{11} T^{2} \) |
| 79 | \( 1 + 3720542360 T + p^{11} T^{2} \) |
| 83 | \( 1 - 8768454036 T + p^{11} T^{2} \) |
| 89 | \( 1 - 25472769174 T + p^{11} T^{2} \) |
| 97 | \( 1 + 39092494846 T + p^{11} T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.18222946685337506230798680590, −9.809344167726347759083905287565, −9.128431896912291058623880729557, −7.931166512902977902080104877504, −6.69161076041514730394750614354, −5.71543777085533730530041854623, −4.55833494924166046283020524051, −3.24833218879637676613124118952, −1.79450540195691325017200641961, −0.973635053188638942623286276921,
0.973635053188638942623286276921, 1.79450540195691325017200641961, 3.24833218879637676613124118952, 4.55833494924166046283020524051, 5.71543777085533730530041854623, 6.69161076041514730394750614354, 7.931166512902977902080104877504, 9.128431896912291058623880729557, 9.809344167726347759083905287565, 11.18222946685337506230798680590
