| L(s) = 1 | − 3-s + 5-s − 5·7-s − 2·9-s − 13-s − 15-s + 3·17-s − 2·19-s + 5·21-s + 4·23-s − 4·25-s + 5·27-s − 6·29-s − 4·31-s − 5·35-s − 11·37-s + 39-s − 8·41-s − 43-s − 2·45-s + 9·47-s + 18·49-s − 3·51-s + 12·53-s + 2·57-s − 6·59-s + 10·63-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s − 1.88·7-s − 2/3·9-s − 0.277·13-s − 0.258·15-s + 0.727·17-s − 0.458·19-s + 1.09·21-s + 0.834·23-s − 4/5·25-s + 0.962·27-s − 1.11·29-s − 0.718·31-s − 0.845·35-s − 1.80·37-s + 0.160·39-s − 1.24·41-s − 0.152·43-s − 0.298·45-s + 1.31·47-s + 18/7·49-s − 0.420·51-s + 1.64·53-s + 0.264·57-s − 0.781·59-s + 1.25·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 100672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 11 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 11 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 + 6 T + p T^{2} \) |
| 71 | \( 1 - 7 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−14.04014673618574, −13.73409936863778, −13.20479840750789, −12.73450777786745, −12.29780495006127, −11.93941943771516, −11.31370564121289, −10.68533902466984, −10.29603737907591, −9.848338812727539, −9.388948501585325, −8.793536487004692, −8.533676636049661, −7.496377820901091, −7.030632538060711, −6.730204701694231, −5.973064463667153, −5.573567736021578, −5.435957256699284, −4.444818379695328, −3.644895734750355, −3.343010980515842, −2.689517471809472, −2.047866491281775, −1.132920644393482, 0, 0,
1.132920644393482, 2.047866491281775, 2.689517471809472, 3.343010980515842, 3.644895734750355, 4.444818379695328, 5.435957256699284, 5.573567736021578, 5.973064463667153, 6.730204701694231, 7.030632538060711, 7.496377820901091, 8.533676636049661, 8.793536487004692, 9.388948501585325, 9.848338812727539, 10.29603737907591, 10.68533902466984, 11.31370564121289, 11.93941943771516, 12.29780495006127, 12.73450777786745, 13.20479840750789, 13.73409936863778, 14.04014673618574
