| L(s) = 1 | − 9-s + 4·11-s − 10·23-s − 2·29-s + 10·37-s + 26·43-s − 4·53-s − 8·67-s + 30·71-s + 2·79-s + 9·81-s − 4·99-s + 38·107-s − 2·109-s − 34·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 4·169-s + 173-s + 179-s + ⋯ |
| L(s) = 1 | − 1/3·9-s + 1.20·11-s − 2.08·23-s − 0.371·29-s + 1.64·37-s + 3.96·43-s − 0.549·53-s − 0.977·67-s + 3.56·71-s + 0.225·79-s + 81-s − 0.402·99-s + 3.67·107-s − 0.191·109-s − 3.09·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 4/13·169-s + 0.0760·173-s + 0.0747·179-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.930925197\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.930925197\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2^3$ | \( 1 + T^{2} - 8 T^{4} + p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $C_2$ | \( ( 1 - T + p T^{2} )^{4} \) |
| 13 | $C_2^2$ | \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^3$ | \( 1 + 29 T^{2} + 552 T^{4} + 29 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 + 5 T^{2} + 492 T^{4} + 5 p^{2} T^{6} + p^{4} T^{8} \) |
| 23 | $D_{4}$ | \( ( 1 + 5 T + 26 T^{2} + 5 p T^{3} + p^{2} T^{4} )^{2} \) |
| 29 | $D_{4}$ | \( ( 1 + T + 32 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 + 80 T^{2} + 3102 T^{4} + 80 p^{2} T^{6} + p^{4} T^{8} \) |
| 37 | $D_{4}$ | \( ( 1 - 5 T + 54 T^{2} - 5 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $D_4\times C_2$ | \( 1 + 65 T^{2} + 2292 T^{4} + 65 p^{2} T^{6} + p^{4} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 - 13 T + 102 T^{2} - 13 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 + 144 T^{2} + 9182 T^{4} + 144 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $C_2^2$ | \( ( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + 90 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 + 62 T^{2} + p^{2} T^{4} )^{2} \) |
| 67 | $D_{4}$ | \( ( 1 + 4 T + 33 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $D_{4}$ | \( ( 1 - 15 T + 172 T^{2} - 15 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 + 17 T^{2} - 5676 T^{4} + 17 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 - T + 132 T^{2} - p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 + 177 T^{2} + 15704 T^{4} + 177 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 + 201 T^{2} + 20036 T^{4} + 201 p^{2} T^{6} + p^{4} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 + 104 T^{2} + 17742 T^{4} + 104 p^{2} T^{6} + p^{4} T^{8} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−5.54101369699747572966027452940, −5.12578658037366511870971588217, −5.08656020630590557996752805308, −4.83938638200591550988211342113, −4.45853351333583315919584417871, −4.41865405007240914182988856263, −4.24405412281982370960610721113, −4.16353352128115600372429473003, −3.96510893904991235973868230525, −3.72429811257568522741361783376, −3.52597137777758898646553794001, −3.37747498137299503048281685892, −3.21125859091264905195727975310, −2.94576721109856125306702078976, −2.54354510713419792098326882296, −2.39248770770020667438062764603, −2.33457641683640967671578660033, −2.16404826108022687946621151763, −1.95430070246354500241172629491, −1.48721739684222064034241394804, −1.39790023606428485914907811540, −1.04849996911517172126367969196, −0.833741835833003221157467211321, −0.46551528938152126918754803684, −0.38414409146716909912625262272,
0.38414409146716909912625262272, 0.46551528938152126918754803684, 0.833741835833003221157467211321, 1.04849996911517172126367969196, 1.39790023606428485914907811540, 1.48721739684222064034241394804, 1.95430070246354500241172629491, 2.16404826108022687946621151763, 2.33457641683640967671578660033, 2.39248770770020667438062764603, 2.54354510713419792098326882296, 2.94576721109856125306702078976, 3.21125859091264905195727975310, 3.37747498137299503048281685892, 3.52597137777758898646553794001, 3.72429811257568522741361783376, 3.96510893904991235973868230525, 4.16353352128115600372429473003, 4.24405412281982370960610721113, 4.41865405007240914182988856263, 4.45853351333583315919584417871, 4.83938638200591550988211342113, 5.08656020630590557996752805308, 5.12578658037366511870971588217, 5.54101369699747572966027452940
