| L(s) = 1 | − 1.59e6·3-s + 1.85e11·7-s + 2.54e12·9-s − 3.15e14·13-s − 3.04e16·19-s − 2.95e17·21-s + 1.49e18·25-s − 4.05e18·27-s + 4.84e19·31-s − 4.86e20·37-s + 5.02e20·39-s + 7.54e20·43-s + 2.50e22·49-s + 4.86e22·57-s + 2.63e23·61-s + 4.71e23·63-s + 8.10e23·67-s + 2.85e24·73-s − 2.37e24·75-s − 8.66e24·79-s + 6.46e24·81-s − 5.84e25·91-s − 7.72e25·93-s + 1.79e25·97-s − 7.36e25·103-s + 4.89e26·109-s + 7.75e26·111-s + ⋯ |
| L(s) = 1 | − 3-s + 1.91·7-s + 9-s − 1.04·13-s − 0.725·19-s − 1.91·21-s + 25-s − 27-s + 1.98·31-s − 1.99·37-s + 1.04·39-s + 0.439·43-s + 2.66·49-s + 0.725·57-s + 1.62·61-s + 1.91·63-s + 1.47·67-s + 1.70·73-s − 75-s − 1.85·79-s + 81-s − 1.99·91-s − 1.98·93-s + 0.267·97-s − 0.501·103-s + 1.59·109-s + 1.99·111-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(27-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s+13) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{27}{2})\) |
\(\approx\) |
\(1.781668655\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.781668655\) |
| \(L(14)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + p^{13} T \) |
| good | 5 | \( ( 1 - p^{13} T )( 1 + p^{13} T ) \) |
| 7 | \( 1 - 185535080642 T + p^{26} T^{2} \) |
| 11 | \( ( 1 - p^{13} T )( 1 + p^{13} T ) \) |
| 13 | \( 1 + 315105612101302 T + p^{26} T^{2} \) |
| 17 | \( ( 1 - p^{13} T )( 1 + p^{13} T ) \) |
| 19 | \( 1 + 30491635106368774 T + p^{26} T^{2} \) |
| 23 | \( ( 1 - p^{13} T )( 1 + p^{13} T ) \) |
| 29 | \( ( 1 - p^{13} T )( 1 + p^{13} T ) \) |
| 31 | \( 1 - 48434386391053004114 T + p^{26} T^{2} \) |
| 37 | \( 1 + \)\(48\!\cdots\!94\)\( T + p^{26} T^{2} \) |
| 41 | \( ( 1 - p^{13} T )( 1 + p^{13} T ) \) |
| 43 | \( 1 - \)\(75\!\cdots\!98\)\( T + p^{26} T^{2} \) |
| 47 | \( ( 1 - p^{13} T )( 1 + p^{13} T ) \) |
| 53 | \( ( 1 - p^{13} T )( 1 + p^{13} T ) \) |
| 59 | \( ( 1 - p^{13} T )( 1 + p^{13} T ) \) |
| 61 | \( 1 - \)\(26\!\cdots\!74\)\( T + p^{26} T^{2} \) |
| 67 | \( 1 - \)\(81\!\cdots\!02\)\( T + p^{26} T^{2} \) |
| 71 | \( ( 1 - p^{13} T )( 1 + p^{13} T ) \) |
| 73 | \( 1 - \)\(28\!\cdots\!34\)\( T + p^{26} T^{2} \) |
| 79 | \( 1 + \)\(86\!\cdots\!22\)\( T + p^{26} T^{2} \) |
| 83 | \( ( 1 - p^{13} T )( 1 + p^{13} T ) \) |
| 89 | \( ( 1 - p^{13} T )( 1 + p^{13} T ) \) |
| 97 | \( 1 - \)\(17\!\cdots\!42\)\( T + p^{26} T^{2} \) |
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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
−14.20253093728105838806777800492, −12.37468276751900472230860446951, −11.37759402789199772555160860585, −10.30204305513813465760735582409, −8.328289668825999528646278272239, −6.96842359237404149449810521937, −5.24891534720719551557627197300, −4.48408481737607948609430464191, −2.05671777661465266331604503554, −0.813016178332185439707090616033,
0.813016178332185439707090616033, 2.05671777661465266331604503554, 4.48408481737607948609430464191, 5.24891534720719551557627197300, 6.96842359237404149449810521937, 8.328289668825999528646278272239, 10.30204305513813465760735582409, 11.37759402789199772555160860585, 12.37468276751900472230860446951, 14.20253093728105838806777800492
