| L(s) = 1 | − 2-s + 4-s − 3·5-s + 7-s − 8-s + 3·10-s + 4·13-s − 14-s + 16-s − 2·19-s − 3·20-s + 6·23-s + 4·25-s − 4·26-s + 28-s + 6·29-s + 5·31-s − 32-s − 3·35-s + 2·37-s + 2·38-s + 3·40-s − 6·41-s + 10·43-s − 6·46-s − 6·47-s − 6·49-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 1.34·5-s + 0.377·7-s − 0.353·8-s + 0.948·10-s + 1.10·13-s − 0.267·14-s + 1/4·16-s − 0.458·19-s − 0.670·20-s + 1.25·23-s + 4/5·25-s − 0.784·26-s + 0.188·28-s + 1.11·29-s + 0.898·31-s − 0.176·32-s − 0.507·35-s + 0.328·37-s + 0.324·38-s + 0.474·40-s − 0.937·41-s + 1.52·43-s − 0.884·46-s − 0.875·47-s − 6/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6534 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6534 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.099156509\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.099156509\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 11 | \( 1 \) |
| good | 5 | \( 1 + 3 T + p T^{2} \) |
| 7 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 14 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 7 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 3 T + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 97 | \( 1 + T + p T^{2} \) |
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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
−8.042628745182811161935615746352, −7.58295634767937201649560498327, −6.64744871264175511687502868338, −6.22717720144741613250664024700, −4.99276705781856711264368753226, −4.39362202012594373576625616984, −3.49004127209409597331438255283, −2.84374221243203227577749155458, −1.53162989918637080882193794820, −0.63937536876372413000711819123,
0.63937536876372413000711819123, 1.53162989918637080882193794820, 2.84374221243203227577749155458, 3.49004127209409597331438255283, 4.39362202012594373576625616984, 4.99276705781856711264368753226, 6.22717720144741613250664024700, 6.64744871264175511687502868338, 7.58295634767937201649560498327, 8.042628745182811161935615746352
