| L(s) = 1 | − 2·3-s − 2·5-s + 2·7-s + 9-s + 4·15-s − 12·17-s − 4·21-s + 3·25-s + 4·27-s − 4·35-s + 4·37-s + 12·41-s − 20·43-s − 2·45-s − 12·47-s − 3·49-s + 24·51-s + 24·59-s + 2·63-s + 4·67-s − 6·75-s + 16·79-s − 11·81-s + 12·83-s + 24·85-s − 12·89-s + 12·101-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 0.894·5-s + 0.755·7-s + 1/3·9-s + 1.03·15-s − 2.91·17-s − 0.872·21-s + 3/5·25-s + 0.769·27-s − 0.676·35-s + 0.657·37-s + 1.87·41-s − 3.04·43-s − 0.298·45-s − 1.75·47-s − 3/7·49-s + 3.36·51-s + 3.12·59-s + 0.251·63-s + 0.488·67-s − 0.692·75-s + 1.80·79-s − 1.22·81-s + 1.31·83-s + 2.60·85-s − 1.27·89-s + 1.19·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 176400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 176400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5802482761\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5802482761\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 7 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| good | 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 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
−9.102954926132022140189127164955, −8.552217550781204179646223792184, −8.257961828526166381407774133620, −7.84705746007080707747306683679, −7.02154360198695631694990895041, −6.57891116465648258947670054106, −6.53418433426898175298075758337, −5.64205532362515499365982008548, −4.94497854065074162838342241506, −4.78130792717525308450176413839, −4.20748366665210762546679649438, −3.57122359553730834302448832528, −2.57999868391564120109847897545, −1.85149647197241415984585908916, −0.52100658180081998819004041300,
0.52100658180081998819004041300, 1.85149647197241415984585908916, 2.57999868391564120109847897545, 3.57122359553730834302448832528, 4.20748366665210762546679649438, 4.78130792717525308450176413839, 4.94497854065074162838342241506, 5.64205532362515499365982008548, 6.53418433426898175298075758337, 6.57891116465648258947670054106, 7.02154360198695631694990895041, 7.84705746007080707747306683679, 8.257961828526166381407774133620, 8.552217550781204179646223792184, 9.102954926132022140189127164955
