| L(s) = 1 | − 1.37e11·2-s − 6.83e15·3-s + 1.88e22·4-s + 9.40e26·6-s − 2.59e33·8-s − 2.02e35·9-s + 2.57e38·11-s − 1.29e38·12-s + 3.56e44·16-s + 5.52e45·17-s + 2.78e46·18-s + 6.13e46·19-s − 3.54e49·22-s + 1.77e49·24-s + 5.29e51·25-s + 2.77e51·27-s − 4.90e55·32-s − 1.76e54·33-s − 7.59e56·34-s − 3.82e57·36-s − 8.43e57·38-s + 4.21e58·41-s − 1.26e60·43-s + 4.87e60·44-s − 2.44e60·48-s + 3.44e62·49-s − 7.27e62·50-s + ⋯ |
| L(s) = 1 | − 2-s − 0.0151·3-s + 4-s + 0.0151·6-s − 8-s − 0.999·9-s + 0.758·11-s − 0.0151·12-s + 16-s + 1.64·17-s + 0.999·18-s + 0.297·19-s − 0.758·22-s + 0.0151·24-s + 25-s + 0.0303·27-s − 32-s − 0.0115·33-s − 1.64·34-s − 0.999·36-s − 0.297·38-s + 0.0893·41-s − 0.461·43-s + 0.758·44-s − 0.0151·48-s + 49-s − 50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(75-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s+37) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{75}{2})\) |
\(\approx\) |
\(1.494666023\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.494666023\) |
| \(L(38)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p^{37} T \) |
| good | 3 | \( 1 + 6839778690897122 T + p^{74} T^{2} \) |
| 5 | \( ( 1 - p^{37} T )( 1 + p^{37} T ) \) |
| 7 | \( ( 1 - p^{37} T )( 1 + p^{37} T ) \) |
| 11 | \( 1 - \)\(25\!\cdots\!54\)\( T + p^{74} T^{2} \) |
| 13 | \( ( 1 - p^{37} T )( 1 + p^{37} T ) \) |
| 17 | \( 1 - \)\(55\!\cdots\!22\)\( T + p^{74} T^{2} \) |
| 19 | \( 1 - \)\(61\!\cdots\!06\)\( T + p^{74} T^{2} \) |
| 23 | \( ( 1 - p^{37} T )( 1 + p^{37} T ) \) |
| 29 | \( ( 1 - p^{37} T )( 1 + p^{37} T ) \) |
| 31 | \( ( 1 - p^{37} T )( 1 + p^{37} T ) \) |
| 37 | \( ( 1 - p^{37} T )( 1 + p^{37} T ) \) |
| 41 | \( 1 - \)\(42\!\cdots\!34\)\( T + p^{74} T^{2} \) |
| 43 | \( 1 + \)\(12\!\cdots\!86\)\( T + p^{74} T^{2} \) |
| 47 | \( ( 1 - p^{37} T )( 1 + p^{37} T ) \) |
| 53 | \( ( 1 - p^{37} T )( 1 + p^{37} T ) \) |
| 59 | \( 1 - \)\(10\!\cdots\!38\)\( T + p^{74} T^{2} \) |
| 61 | \( ( 1 - p^{37} T )( 1 + p^{37} T ) \) |
| 67 | \( 1 + \)\(63\!\cdots\!58\)\( T + p^{74} T^{2} \) |
| 71 | \( ( 1 - p^{37} T )( 1 + p^{37} T ) \) |
| 73 | \( 1 - \)\(12\!\cdots\!98\)\( T + p^{74} T^{2} \) |
| 79 | \( ( 1 - p^{37} T )( 1 + p^{37} T ) \) |
| 83 | \( 1 - \)\(10\!\cdots\!98\)\( T + p^{74} T^{2} \) |
| 89 | \( 1 + \)\(22\!\cdots\!34\)\( T + p^{74} T^{2} \) |
| 97 | \( 1 - \)\(86\!\cdots\!26\)\( T + p^{74} T^{2} \) |
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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
−10.43606208459448802359871521947, −9.354198623789585115514035345549, −8.460470675712844474155652348217, −7.43869644882619398415424404006, −6.29312838825837172320271103592, −5.33261738393525472024880150462, −3.54080192689938391846209423195, −2.68630780405226854762673287368, −1.39709123059781409822585294636, −0.60566421139381360178885811258,
0.60566421139381360178885811258, 1.39709123059781409822585294636, 2.68630780405226854762673287368, 3.54080192689938391846209423195, 5.33261738393525472024880150462, 6.29312838825837172320271103592, 7.43869644882619398415424404006, 8.460470675712844474155652348217, 9.354198623789585115514035345549, 10.43606208459448802359871521947
