| L(s) = 1 | − 3·3-s − 7·7-s + 9·9-s + 2·13-s − 22·17-s + 21·21-s − 38·23-s + 25·25-s − 27·27-s + 26·29-s + 34·31-s − 6·39-s + 26·41-s + 82·43-s + 49·49-s + 66·51-s − 22·53-s + 106·59-s − 94·61-s − 63·63-s + 34·67-s + 114·69-s + 58·71-s − 75·75-s + 81·81-s + 58·83-s − 78·87-s + ⋯ |
| L(s) = 1 | − 3-s − 7-s + 9-s + 2/13·13-s − 1.29·17-s + 21-s − 1.65·23-s + 25-s − 27-s + 0.896·29-s + 1.09·31-s − 0.153·39-s + 0.634·41-s + 1.90·43-s + 49-s + 1.29·51-s − 0.415·53-s + 1.79·59-s − 1.54·61-s − 63-s + 0.507·67-s + 1.65·69-s + 0.816·71-s − 75-s + 81-s + 0.698·83-s − 0.896·87-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.9263566330\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9263566330\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + p T \) |
| 7 | \( 1 + p T \) |
| good | 5 | \( ( 1 - p T )( 1 + p T ) \) |
| 11 | \( ( 1 - p T )( 1 + p T ) \) |
| 13 | \( 1 - 2 T + p^{2} T^{2} \) |
| 17 | \( 1 + 22 T + p^{2} T^{2} \) |
| 19 | \( ( 1 - p T )( 1 + p T ) \) |
| 23 | \( 1 + 38 T + p^{2} T^{2} \) |
| 29 | \( 1 - 26 T + p^{2} T^{2} \) |
| 31 | \( 1 - 34 T + p^{2} T^{2} \) |
| 37 | \( ( 1 - p T )( 1 + p T ) \) |
| 41 | \( 1 - 26 T + p^{2} T^{2} \) |
| 43 | \( 1 - 82 T + p^{2} T^{2} \) |
| 47 | \( ( 1 - p T )( 1 + p T ) \) |
| 53 | \( 1 + 22 T + p^{2} T^{2} \) |
| 59 | \( 1 - 106 T + p^{2} T^{2} \) |
| 61 | \( 1 + 94 T + p^{2} T^{2} \) |
| 67 | \( 1 - 34 T + p^{2} T^{2} \) |
| 71 | \( 1 - 58 T + p^{2} T^{2} \) |
| 73 | \( ( 1 - p T )( 1 + p T ) \) |
| 79 | \( ( 1 - p T )( 1 + p T ) \) |
| 83 | \( 1 - 58 T + p^{2} T^{2} \) |
| 89 | \( 1 - 122 T + p^{2} T^{2} \) |
| 97 | \( ( 1 - p T )( 1 + p T ) \) |
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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
−10.37201805440193447172951810202, −9.632543154892364177052882303864, −8.675466776990767693499307260327, −7.46619449369695535850565683601, −6.45625241073579188105822107257, −6.09000507465874906514867043334, −4.78622709428025575038547179850, −3.91666419500503953828492308931, −2.42533697503928388424039992461, −0.67121131489636621110663435335,
0.67121131489636621110663435335, 2.42533697503928388424039992461, 3.91666419500503953828492308931, 4.78622709428025575038547179850, 6.09000507465874906514867043334, 6.45625241073579188105822107257, 7.46619449369695535850565683601, 8.675466776990767693499307260327, 9.632543154892364177052882303864, 10.37201805440193447172951810202
