| L(s) = 1 | − 4·5-s − 6·7-s − 9-s − 2·11-s + 8·13-s + 6·17-s + 6·19-s + 2·23-s + 11·25-s + 6·29-s + 24·35-s + 16·37-s + 12·43-s + 4·45-s − 10·47-s + 18·49-s + 8·55-s + 14·59-s − 18·61-s + 6·63-s − 32·65-s + 4·67-s + 16·71-s − 10·73-s + 12·77-s + 81-s − 24·85-s + ⋯ |
| L(s) = 1 | − 1.78·5-s − 2.26·7-s − 1/3·9-s − 0.603·11-s + 2.21·13-s + 1.45·17-s + 1.37·19-s + 0.417·23-s + 11/5·25-s + 1.11·29-s + 4.05·35-s + 2.63·37-s + 1.82·43-s + 0.596·45-s − 1.45·47-s + 18/7·49-s + 1.07·55-s + 1.82·59-s − 2.30·61-s + 0.755·63-s − 3.96·65-s + 0.488·67-s + 1.89·71-s − 1.17·73-s + 1.36·77-s + 1/9·81-s − 2.60·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 921600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 921600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.248585816\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.248585816\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 5 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 10 T + 50 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 14 T + 98 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 18 T + 162 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 150 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 14 T + 98 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
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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
−10.05666455146787269732823924214, −9.894678045649337563943791647709, −9.409896548029265866693877053678, −8.991645488224489472355898403063, −8.526812315973753380135801107371, −8.080670316971243145538409775595, −7.70936025147659137853429108406, −7.46848789369420650907225442905, −6.74272064537000155325192388997, −6.47966922993428207882949473400, −5.89654118929861915696712341992, −5.74229659407871721222421844457, −4.93322811690000764927892897733, −4.30671189862613375022402353157, −3.72089945689145135046618959406, −3.45924509779825604876518045058, −3.01178230664606519054703495668, −2.77890855266839489798185807409, −0.999524112525943013884931767549, −0.70536687291686079420166310973,
0.70536687291686079420166310973, 0.999524112525943013884931767549, 2.77890855266839489798185807409, 3.01178230664606519054703495668, 3.45924509779825604876518045058, 3.72089945689145135046618959406, 4.30671189862613375022402353157, 4.93322811690000764927892897733, 5.74229659407871721222421844457, 5.89654118929861915696712341992, 6.47966922993428207882949473400, 6.74272064537000155325192388997, 7.46848789369420650907225442905, 7.70936025147659137853429108406, 8.080670316971243145538409775595, 8.526812315973753380135801107371, 8.991645488224489472355898403063, 9.409896548029265866693877053678, 9.894678045649337563943791647709, 10.05666455146787269732823924214
