| L(s) = 1 | − 3-s + 9-s + 11-s − 13-s + 8·17-s − 7·19-s + 2·23-s − 27-s − 29-s − 4·31-s − 33-s + 3·37-s + 39-s − 6·41-s − 4·43-s − 47-s − 8·51-s − 6·53-s + 7·57-s − 3·59-s − 14·61-s + 7·67-s − 2·69-s + 6·71-s − 11·73-s − 4·79-s + 81-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/3·9-s + 0.301·11-s − 0.277·13-s + 1.94·17-s − 1.60·19-s + 0.417·23-s − 0.192·27-s − 0.185·29-s − 0.718·31-s − 0.174·33-s + 0.493·37-s + 0.160·39-s − 0.937·41-s − 0.609·43-s − 0.145·47-s − 1.12·51-s − 0.824·53-s + 0.927·57-s − 0.390·59-s − 1.79·61-s + 0.855·67-s − 0.240·69-s + 0.712·71-s − 1.28·73-s − 0.450·79-s + 1/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 323400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 323400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 - 8 T + p T^{2} \) |
| 19 | \( 1 + 7 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 - 7 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + 11 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 + 16 T + p T^{2} \) |
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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
−12.69628226042464, −12.37783876679992, −12.08162202315668, −11.30526083884117, −11.20091923851545, −10.55610760858055, −10.08425816541342, −9.835003345199076, −9.210862865614960, −8.741056648119643, −8.190345538914217, −7.740548693336062, −7.278917469494764, −6.752960671984124, −6.256797054501842, −5.833230855578192, −5.361234159611226, −4.788903242287174, −4.402188643862300, −3.704565455826775, −3.279334554971398, −2.698525302163181, −1.801470770361831, −1.522679749878172, −0.6891646100858187, 0,
0.6891646100858187, 1.522679749878172, 1.801470770361831, 2.698525302163181, 3.279334554971398, 3.704565455826775, 4.402188643862300, 4.788903242287174, 5.361234159611226, 5.833230855578192, 6.256797054501842, 6.752960671984124, 7.278917469494764, 7.740548693336062, 8.190345538914217, 8.741056648119643, 9.210862865614960, 9.835003345199076, 10.08425816541342, 10.55610760858055, 11.20091923851545, 11.30526083884117, 12.08162202315668, 12.37783876679992, 12.69628226042464
