| L(s) = 1 | + 2·7-s + 4·11-s − 13-s + 6·17-s − 6·19-s − 5·25-s − 2·29-s − 6·31-s − 10·37-s − 8·41-s − 12·43-s − 12·47-s − 3·49-s − 6·53-s − 2·61-s − 2·67-s + 8·71-s + 14·73-s + 8·77-s + 4·79-s + 8·83-s − 4·89-s − 2·91-s + 14·97-s − 18·101-s + 4·103-s − 4·107-s + ⋯ |
| L(s) = 1 | + 0.755·7-s + 1.20·11-s − 0.277·13-s + 1.45·17-s − 1.37·19-s − 25-s − 0.371·29-s − 1.07·31-s − 1.64·37-s − 1.24·41-s − 1.82·43-s − 1.75·47-s − 3/7·49-s − 0.824·53-s − 0.256·61-s − 0.244·67-s + 0.949·71-s + 1.63·73-s + 0.911·77-s + 0.450·79-s + 0.878·83-s − 0.423·89-s − 0.209·91-s + 1.42·97-s − 1.79·101-s + 0.394·103-s − 0.386·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7488 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7488 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 + 4 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−7.64318606322212896793502114540, −6.74885857601312734498011594370, −6.29726020464242302145032155270, −5.26907258940557511614007229475, −4.86255818677025553674986843061, −3.74086617900481051255118226359, −3.42875280862789178740168493737, −1.89609049823861750791828359250, −1.56722806823473287981560740603, 0,
1.56722806823473287981560740603, 1.89609049823861750791828359250, 3.42875280862789178740168493737, 3.74086617900481051255118226359, 4.86255818677025553674986843061, 5.26907258940557511614007229475, 6.29726020464242302145032155270, 6.74885857601312734498011594370, 7.64318606322212896793502114540
