L(s) = 1 | + 3·3-s + 4·4-s − 11·7-s + 9·9-s + 12·12-s + 13-s + 16·16-s − 37·19-s − 33·21-s + 27·27-s − 44·28-s − 13·31-s + 36·36-s − 26·37-s + 3·39-s + 61·43-s + 48·48-s + 72·49-s + 4·52-s − 111·57-s + 47·61-s − 99·63-s + 64·64-s + 109·67-s + 46·73-s − 148·76-s − 142·79-s + ⋯ |
L(s) = 1 | + 3-s + 4-s − 1.57·7-s + 9-s + 12-s + 1/13·13-s + 16-s − 1.94·19-s − 1.57·21-s + 27-s − 1.57·28-s − 0.419·31-s + 36-s − 0.702·37-s + 1/13·39-s + 1.41·43-s + 48-s + 1.46·49-s + 1/13·52-s − 1.94·57-s + 0.770·61-s − 1.57·63-s + 64-s + 1.62·67-s + 0.630·73-s − 1.94·76-s − 1.79·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.752314020\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.752314020\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - p T \) |
| 5 | \( 1 \) |
good | 2 | \( ( 1 - p T )( 1 + p T ) \) |
| 7 | \( 1 + 11 T + p^{2} T^{2} \) |
| 11 | \( ( 1 - p T )( 1 + p T ) \) |
| 13 | \( 1 - T + p^{2} T^{2} \) |
| 17 | \( ( 1 - p T )( 1 + p T ) \) |
| 19 | \( 1 + 37 T + p^{2} T^{2} \) |
| 23 | \( ( 1 - p T )( 1 + p T ) \) |
| 29 | \( ( 1 - p T )( 1 + p T ) \) |
| 31 | \( 1 + 13 T + p^{2} T^{2} \) |
| 37 | \( 1 + 26 T + p^{2} T^{2} \) |
| 41 | \( ( 1 - p T )( 1 + p T ) \) |
| 43 | \( 1 - 61 T + p^{2} T^{2} \) |
| 47 | \( ( 1 - p T )( 1 + p T ) \) |
| 53 | \( ( 1 - p T )( 1 + p T ) \) |
| 59 | \( ( 1 - p T )( 1 + p T ) \) |
| 61 | \( 1 - 47 T + p^{2} T^{2} \) |
| 67 | \( 1 - 109 T + p^{2} T^{2} \) |
| 71 | \( ( 1 - p T )( 1 + p T ) \) |
| 73 | \( 1 - 46 T + p^{2} T^{2} \) |
| 79 | \( 1 + 142 T + p^{2} T^{2} \) |
| 83 | \( ( 1 - p T )( 1 + p T ) \) |
| 89 | \( ( 1 - p T )( 1 + p T ) \) |
| 97 | \( 1 - 169 T + p^{2} T^{2} \) |
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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
−14.42707443017962127893503757179, −13.04667991370884111735625737301, −12.46213701794879877616905149124, −10.76432070397562879502361196424, −9.808637396540701377706701613764, −8.590648228047442639180407067464, −7.15891220772788699271069501067, −6.25460592902984727996751972766, −3.72576451583680290568787903805, −2.41569492732756435213236695237,
2.41569492732756435213236695237, 3.72576451583680290568787903805, 6.25460592902984727996751972766, 7.15891220772788699271069501067, 8.590648228047442639180407067464, 9.808637396540701377706701613764, 10.76432070397562879502361196424, 12.46213701794879877616905149124, 13.04667991370884111735625737301, 14.42707443017962127893503757179