| L(s) = 1 | + 2·3-s + 4-s − 3·9-s − 3·11-s + 2·12-s + 16-s − 6·23-s − 10·25-s − 14·27-s + 10·31-s − 6·33-s − 3·36-s − 14·37-s − 3·44-s − 6·47-s + 2·48-s + 49-s − 24·53-s + 12·59-s + 64-s + 10·67-s − 12·69-s + 24·71-s − 20·75-s − 4·81-s − 36·89-s − 6·92-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 1/2·4-s − 9-s − 0.904·11-s + 0.577·12-s + 1/4·16-s − 1.25·23-s − 2·25-s − 2.69·27-s + 1.79·31-s − 1.04·33-s − 1/2·36-s − 2.30·37-s − 0.452·44-s − 0.875·47-s + 0.288·48-s + 1/7·49-s − 3.29·53-s + 1.56·59-s + 1/8·64-s + 1.22·67-s − 1.44·69-s + 2.84·71-s − 2.30·75-s − 4/9·81-s − 3.81·89-s − 0.625·92-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4008004 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4008004 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.391677390\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.391677390\) |
| \(L(\frac{3}{2})\) |
|
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$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 11 | $C_2$ | \( 1 + 3 T + p T^{2} \) |
| 13 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 3 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + 18 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 17 T + p T^{2} )^{2} \) |
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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
−7.63380170955425435565378568582, −7.12452515455087807112624959383, −6.54090950323955775391739886563, −6.10743185313712551583625613621, −5.94928285901447704557672794095, −5.27901795354844706811221007129, −5.06509234546123458101387314677, −4.41584211850724569307269915633, −3.60858467418748472558474969025, −3.53538503374708697062546960940, −3.07464268547435599479371512552, −2.33000973337707432224237511883, −2.22414801159896403461802083456, −1.68074999787226013079846229266, −0.35275945748628279067689817923,
0.35275945748628279067689817923, 1.68074999787226013079846229266, 2.22414801159896403461802083456, 2.33000973337707432224237511883, 3.07464268547435599479371512552, 3.53538503374708697062546960940, 3.60858467418748472558474969025, 4.41584211850724569307269915633, 5.06509234546123458101387314677, 5.27901795354844706811221007129, 5.94928285901447704557672794095, 6.10743185313712551583625613621, 6.54090950323955775391739886563, 7.12452515455087807112624959383, 7.63380170955425435565378568582
