| L(s) = 1 | − 1.16e9·3-s − 9.25e15·7-s + 1.35e18·9-s + 9.09e20·13-s − 3.89e24·19-s + 1.07e25·21-s + 3.63e26·25-s − 1.57e27·27-s − 7.55e27·31-s − 6.40e29·37-s − 1.05e30·39-s − 2.12e31·43-s − 4.42e31·49-s + 4.52e33·57-s + 3.04e33·61-s − 1.25e34·63-s − 2.43e34·67-s − 9.71e34·73-s − 4.22e35·75-s + 1.58e36·79-s + 1.82e36·81-s − 8.41e36·91-s + 8.77e36·93-s − 2.18e37·97-s + 3.41e38·103-s + 9.00e38·109-s + 7.44e38·111-s + ⋯ |
| L(s) = 1 | − 3-s − 0.811·7-s + 9-s + 0.622·13-s − 1.96·19-s + 0.811·21-s + 25-s − 27-s − 0.348·31-s − 1.02·37-s − 0.622·39-s − 1.95·43-s − 0.340·49-s + 1.96·57-s + 0.365·61-s − 0.811·63-s − 0.491·67-s − 0.383·73-s − 75-s + 1.39·79-s + 81-s − 0.504·91-s + 0.348·93-s − 0.389·97-s + 1.94·103-s + 1.75·109-s + 1.02·111-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(39-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s+19) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{39}{2})\) |
\(\approx\) |
\(0.8198828688\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8198828688\) |
| \(L(20)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + p^{19} T \) |
| good | 5 | \( ( 1 - p^{19} T )( 1 + p^{19} T ) \) |
| 7 | \( 1 + 9254153518200862 T + p^{38} T^{2} \) |
| 11 | \( ( 1 - p^{19} T )( 1 + p^{19} T ) \) |
| 13 | \( 1 - \)\(90\!\cdots\!42\)\( T + p^{38} T^{2} \) |
| 17 | \( ( 1 - p^{19} T )( 1 + p^{19} T ) \) |
| 19 | \( 1 + \)\(38\!\cdots\!74\)\( T + p^{38} T^{2} \) |
| 23 | \( ( 1 - p^{19} T )( 1 + p^{19} T ) \) |
| 29 | \( ( 1 - p^{19} T )( 1 + p^{19} T ) \) |
| 31 | \( 1 + \)\(75\!\cdots\!06\)\( T + p^{38} T^{2} \) |
| 37 | \( 1 + \)\(64\!\cdots\!46\)\( T + p^{38} T^{2} \) |
| 41 | \( ( 1 - p^{19} T )( 1 + p^{19} T ) \) |
| 43 | \( 1 + \)\(21\!\cdots\!58\)\( T + p^{38} T^{2} \) |
| 47 | \( ( 1 - p^{19} T )( 1 + p^{19} T ) \) |
| 53 | \( ( 1 - p^{19} T )( 1 + p^{19} T ) \) |
| 59 | \( ( 1 - p^{19} T )( 1 + p^{19} T ) \) |
| 61 | \( 1 - \)\(30\!\cdots\!74\)\( T + p^{38} T^{2} \) |
| 67 | \( 1 + \)\(24\!\cdots\!42\)\( T + p^{38} T^{2} \) |
| 71 | \( ( 1 - p^{19} T )( 1 + p^{19} T ) \) |
| 73 | \( 1 + \)\(97\!\cdots\!74\)\( T + p^{38} T^{2} \) |
| 79 | \( 1 - \)\(15\!\cdots\!38\)\( T + p^{38} T^{2} \) |
| 83 | \( ( 1 - p^{19} T )( 1 + p^{19} T ) \) |
| 89 | \( ( 1 - p^{19} T )( 1 + p^{19} T ) \) |
| 97 | \( 1 + \)\(21\!\cdots\!62\)\( T + p^{38} T^{2} \) |
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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
−12.41861442603375978491264083572, −11.04470705702238044234303770037, −10.11727636019149972195938585091, −8.640227973834656627587593505849, −6.87340906411161216447352213209, −6.12039376878981883612171722900, −4.76069085406159247964634445886, −3.50155975065760116766797540626, −1.81675946867034085836316024294, −0.44671266353101894073330131618,
0.44671266353101894073330131618, 1.81675946867034085836316024294, 3.50155975065760116766797540626, 4.76069085406159247964634445886, 6.12039376878981883612171722900, 6.87340906411161216447352213209, 8.640227973834656627587593505849, 10.11727636019149972195938585091, 11.04470705702238044234303770037, 12.41861442603375978491264083572
