| L(s) = 1 | − 2·5-s + 4·7-s − 2·9-s + 4·13-s + 12·23-s + 3·25-s + 6·29-s − 8·35-s + 4·45-s − 2·49-s − 12·53-s + 24·59-s − 8·63-s − 8·65-s + 4·67-s − 24·71-s − 5·81-s + 12·83-s + 16·91-s + 28·103-s − 12·107-s + 4·109-s − 24·115-s − 8·117-s − 22·121-s − 4·125-s + 127-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 1.51·7-s − 2/3·9-s + 1.10·13-s + 2.50·23-s + 3/5·25-s + 1.11·29-s − 1.35·35-s + 0.596·45-s − 2/7·49-s − 1.64·53-s + 3.12·59-s − 1.00·63-s − 0.992·65-s + 0.488·67-s − 2.84·71-s − 5/9·81-s + 1.31·83-s + 1.67·91-s + 2.75·103-s − 1.16·107-s + 0.383·109-s − 2.23·115-s − 0.739·117-s − 2·121-s − 0.357·125-s + 0.0887·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 336400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.975079124\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.975079124\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 29 | $C_2$ | \( 1 - 6 T + p T^{2} \) |
| good | 3 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.552217550781204179646223792184, −8.483197285337311476714919929487, −7.84776053423609074264626236578, −7.54682968289187724168261985660, −6.85164916278026696992828299608, −6.57891116465648258947670054106, −5.83282239676676345421736207728, −5.28343034459676936825398261669, −4.78130792717525308450176413839, −4.54301159938977421825721297318, −3.71708582474438105612348859112, −3.19902910472993221395422173688, −2.62854536549803935810645982732, −1.56433382981290291311865710327, −0.892682357417418065980839508444,
0.892682357417418065980839508444, 1.56433382981290291311865710327, 2.62854536549803935810645982732, 3.19902910472993221395422173688, 3.71708582474438105612348859112, 4.54301159938977421825721297318, 4.78130792717525308450176413839, 5.28343034459676936825398261669, 5.83282239676676345421736207728, 6.57891116465648258947670054106, 6.85164916278026696992828299608, 7.54682968289187724168261985660, 7.84776053423609074264626236578, 8.483197285337311476714919929487, 8.552217550781204179646223792184
