| L(s) = 1 | + 0.539·3-s − 3.70·7-s − 2.70·9-s + 5.51·11-s − 13-s + 0.340·17-s − 2.24·19-s − 2·21-s − 1.46·23-s − 3.07·27-s + 0.630·29-s − 6.58·31-s + 2.97·33-s − 5.55·37-s − 0.539·39-s − 4.68·41-s − 10.6·43-s − 4.29·47-s + 6.75·49-s + 0.183·51-s − 9.07·53-s − 1.21·57-s + 0.986·59-s − 12.2·61-s + 10.0·63-s − 4.97·67-s − 0.787·69-s + ⋯ |
| L(s) = 1 | + 0.311·3-s − 1.40·7-s − 0.903·9-s + 1.66·11-s − 0.277·13-s + 0.0825·17-s − 0.515·19-s − 0.436·21-s − 0.304·23-s − 0.592·27-s + 0.117·29-s − 1.18·31-s + 0.517·33-s − 0.912·37-s − 0.0863·39-s − 0.730·41-s − 1.62·43-s − 0.625·47-s + 0.965·49-s + 0.0256·51-s − 1.24·53-s − 0.160·57-s + 0.128·59-s − 1.56·61-s + 1.26·63-s − 0.607·67-s − 0.0948·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 3 | \( 1 - 0.539T + 3T^{2} \) |
| 7 | \( 1 + 3.70T + 7T^{2} \) |
| 11 | \( 1 - 5.51T + 11T^{2} \) |
| 17 | \( 1 - 0.340T + 17T^{2} \) |
| 19 | \( 1 + 2.24T + 19T^{2} \) |
| 23 | \( 1 + 1.46T + 23T^{2} \) |
| 29 | \( 1 - 0.630T + 29T^{2} \) |
| 31 | \( 1 + 6.58T + 31T^{2} \) |
| 37 | \( 1 + 5.55T + 37T^{2} \) |
| 41 | \( 1 + 4.68T + 41T^{2} \) |
| 43 | \( 1 + 10.6T + 43T^{2} \) |
| 47 | \( 1 + 4.29T + 47T^{2} \) |
| 53 | \( 1 + 9.07T + 53T^{2} \) |
| 59 | \( 1 - 0.986T + 59T^{2} \) |
| 61 | \( 1 + 12.2T + 61T^{2} \) |
| 67 | \( 1 + 4.97T + 67T^{2} \) |
| 71 | \( 1 - 7.32T + 71T^{2} \) |
| 73 | \( 1 + 11.1T + 73T^{2} \) |
| 79 | \( 1 - 9.75T + 79T^{2} \) |
| 83 | \( 1 - 16.8T + 83T^{2} \) |
| 89 | \( 1 - 14.9T + 89T^{2} \) |
| 97 | \( 1 - 12.4T + 97T^{2} \) |
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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
−9.174609680835756039729318141655, −8.716238388292375841070989973581, −7.59885817406925495931237621974, −6.48347232774616470815248654334, −6.28424462363203038473434497472, −5.00489540608203316767806392722, −3.67392056531936646136853240194, −3.24561750650815173183162945043, −1.85570148437830006657774473685, 0,
1.85570148437830006657774473685, 3.24561750650815173183162945043, 3.67392056531936646136853240194, 5.00489540608203316767806392722, 6.28424462363203038473434497472, 6.48347232774616470815248654334, 7.59885817406925495931237621974, 8.716238388292375841070989973581, 9.174609680835756039729318141655
