| L(s) = 1 | + 3.13e10·3-s + 7.81e18·7-s + 9.84e20·9-s + 6.29e24·13-s + 1.75e28·19-s + 2.45e29·21-s + 5.68e30·25-s + 3.09e31·27-s − 1.15e33·31-s + 2.80e33·37-s + 1.97e35·39-s − 1.43e36·43-s + 4.58e37·49-s + 5.49e38·57-s + 7.55e38·61-s + 7.69e39·63-s − 2.98e40·67-s − 1.41e41·73-s + 1.78e41·75-s − 9.47e41·79-s + 9.69e41·81-s + 4.92e43·91-s − 3.63e43·93-s − 9.97e43·97-s + 1.16e44·103-s − 6.28e44·109-s + 8.80e43·111-s + ⋯ |
| L(s) = 1 | + 3-s + 1.99·7-s + 9-s + 1.95·13-s + 1.29·19-s + 1.99·21-s + 25-s + 27-s − 1.79·31-s + 0.0886·37-s + 1.95·39-s − 1.65·43-s + 2.99·49-s + 1.29·57-s + 0.398·61-s + 1.99·63-s − 1.99·67-s − 1.43·73-s + 75-s − 1.69·79-s + 81-s + 3.91·91-s − 1.79·93-s − 1.94·97-s + 0.605·103-s − 0.943·109-s + 0.0886·111-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(45-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s+22) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{45}{2})\) |
\(\approx\) |
\(5.691667127\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.691667127\) |
| \(L(23)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - p^{22} T \) |
| good | 5 | \( ( 1 - p^{22} T )( 1 + p^{22} T ) \) |
| 7 | \( 1 - 7819074024260706146 T + p^{44} T^{2} \) |
| 11 | \( ( 1 - p^{22} T )( 1 + p^{22} T ) \) |
| 13 | \( 1 - \)\(62\!\cdots\!66\)\( T + p^{44} T^{2} \) |
| 17 | \( ( 1 - p^{22} T )( 1 + p^{22} T ) \) |
| 19 | \( 1 - \)\(17\!\cdots\!74\)\( T + p^{44} T^{2} \) |
| 23 | \( ( 1 - p^{22} T )( 1 + p^{22} T ) \) |
| 29 | \( ( 1 - p^{22} T )( 1 + p^{22} T ) \) |
| 31 | \( 1 + \)\(11\!\cdots\!66\)\( T + p^{44} T^{2} \) |
| 37 | \( 1 - \)\(28\!\cdots\!38\)\( T + p^{44} T^{2} \) |
| 41 | \( ( 1 - p^{22} T )( 1 + p^{22} T ) \) |
| 43 | \( 1 + \)\(14\!\cdots\!34\)\( T + p^{44} T^{2} \) |
| 47 | \( ( 1 - p^{22} T )( 1 + p^{22} T ) \) |
| 53 | \( ( 1 - p^{22} T )( 1 + p^{22} T ) \) |
| 59 | \( ( 1 - p^{22} T )( 1 + p^{22} T ) \) |
| 61 | \( 1 - \)\(75\!\cdots\!74\)\( T + p^{44} T^{2} \) |
| 67 | \( 1 + \)\(29\!\cdots\!34\)\( T + p^{44} T^{2} \) |
| 71 | \( ( 1 - p^{22} T )( 1 + p^{22} T ) \) |
| 73 | \( 1 + \)\(14\!\cdots\!42\)\( T + p^{44} T^{2} \) |
| 79 | \( 1 + \)\(94\!\cdots\!18\)\( T + p^{44} T^{2} \) |
| 83 | \( ( 1 - p^{22} T )( 1 + p^{22} T ) \) |
| 89 | \( ( 1 - p^{22} T )( 1 + p^{22} T ) \) |
| 97 | \( 1 + \)\(99\!\cdots\!54\)\( T + p^{44} T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.58464176009624517471833442766, −10.64610819917341854685741838193, −8.942455841327455550042053806696, −8.256700715578192100105126493041, −7.24274898428600835343986814708, −5.44824809947101417792734005804, −4.27373879616656418094670043439, −3.17991263536606569761270084772, −1.65516565292104797417682323659, −1.22887093633112588238111538656,
1.22887093633112588238111538656, 1.65516565292104797417682323659, 3.17991263536606569761270084772, 4.27373879616656418094670043439, 5.44824809947101417792734005804, 7.24274898428600835343986814708, 8.256700715578192100105126493041, 8.942455841327455550042053806696, 10.64610819917341854685741838193, 11.58464176009624517471833442766
