| L(s) = 1 | + 2·2-s + 3·4-s + 4·8-s − 9-s + 8·13-s + 5·16-s + 8·17-s − 2·18-s + 16·26-s + 6·32-s + 16·34-s − 3·36-s − 12·43-s + 16·47-s + 10·49-s + 24·52-s − 2·53-s − 10·59-s + 7·64-s − 24·67-s + 24·68-s − 4·72-s + 81-s + 18·83-s − 24·86-s + 32·94-s + 20·98-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 3/2·4-s + 1.41·8-s − 1/3·9-s + 2.21·13-s + 5/4·16-s + 1.94·17-s − 0.471·18-s + 3.13·26-s + 1.06·32-s + 2.74·34-s − 1/2·36-s − 1.82·43-s + 2.33·47-s + 10/7·49-s + 3.32·52-s − 0.274·53-s − 1.30·59-s + 7/8·64-s − 2.93·67-s + 2.91·68-s − 0.471·72-s + 1/9·81-s + 1.97·83-s − 2.58·86-s + 3.30·94-s + 2.02·98-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6502500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6502500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(9.021903495\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.021903495\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 3 | $C_2$ | \( 1 + T^{2} \) |
| 5 | | \( 1 \) |
| 17 | $C_2$ | \( 1 - 8 T + p T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 45 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 65 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 57 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 - 97 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 + 83 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 142 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
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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
−8.930014477026616024178859723879, −8.577600693067573933850373984039, −8.440662630562411092017997033955, −7.69569014888196136310901126849, −7.51113757926563833653367007706, −7.26193381536280507326984298497, −6.46800566257941483552658679411, −6.28489541361012878894295276832, −5.80640349039043853691066617828, −5.76923159764874239951114011747, −5.20690766666806973011547579825, −4.77370578520073683081716035442, −4.24766301303393162438789064102, −3.80613010147944835317695320457, −3.45098902689483106273865289495, −3.16535851022350304776503013011, −2.67503728252287884576828577808, −1.89304459896753367238661748007, −1.37683404620162526525375947638, −0.851638379595926757998904876692,
0.851638379595926757998904876692, 1.37683404620162526525375947638, 1.89304459896753367238661748007, 2.67503728252287884576828577808, 3.16535851022350304776503013011, 3.45098902689483106273865289495, 3.80613010147944835317695320457, 4.24766301303393162438789064102, 4.77370578520073683081716035442, 5.20690766666806973011547579825, 5.76923159764874239951114011747, 5.80640349039043853691066617828, 6.28489541361012878894295276832, 6.46800566257941483552658679411, 7.26193381536280507326984298497, 7.51113757926563833653367007706, 7.69569014888196136310901126849, 8.440662630562411092017997033955, 8.577600693067573933850373984039, 8.930014477026616024178859723879
