| L(s) = 1 | + 24·17-s + 28·25-s − 216·41-s + 100·49-s − 328·73-s + 456·89-s + 136·97-s + 312·113-s + 380·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 284·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯ |
| L(s) = 1 | + 1.41·17-s + 1.11·25-s − 5.26·41-s + 2.04·49-s − 4.49·73-s + 5.12·89-s + 1.40·97-s + 2.76·113-s + 3.14·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + 1.68·169-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + 0.00518·193-s + 0.00507·197-s + 0.00502·199-s + 0.00473·211-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(4.500674496\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.500674496\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| good | 5 | $C_2$ | \( ( 1 - 8 T + p^{2} T^{2} )^{2}( 1 + 8 T + p^{2} T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 - 50 T^{2} + p^{4} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 190 T^{2} + p^{4} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 142 T^{2} + p^{4} T^{4} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p^{2} T^{2} )^{4} \) |
| 19 | $C_2^2$ | \( ( 1 + 674 T^{2} + p^{4} T^{4} )^{2} \) |
| 23 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 29 | $C_2^2$ | \( ( 1 - 782 T^{2} + p^{4} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 - 1490 T^{2} + p^{4} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 - 2062 T^{2} + p^{4} T^{4} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 54 T + p^{2} T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 + 3266 T^{2} + p^{4} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 - 2690 T^{2} + p^{4} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 5294 T^{2} + p^{4} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + 6530 T^{2} + p^{4} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 - 2542 T^{2} + p^{4} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 - 4894 T^{2} + p^{4} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - 3170 T^{2} + p^{4} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 82 T + p^{2} T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 - 6674 T^{2} + p^{4} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + 13346 T^{2} + p^{4} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 114 T + p^{2} T^{2} )^{4} \) |
| 97 | $C_2$ | \( ( 1 - 34 T + p^{2} T^{2} )^{4} \) |
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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
−6.25863523594975217737596065231, −5.80728262334941218987575966273, −5.78417763685452092199605828084, −5.64049901288981774754337544875, −5.51398904516286972249898819605, −4.97151904127886520704910668594, −4.93798733840619126917019149366, −4.86134871287296512603279605489, −4.57687490804278990761073451145, −4.26784528356735846969131880642, −4.23544502250961265704144209081, −3.59474957724721315819455328579, −3.56735951969467924482382384895, −3.21091273383519435551855237727, −3.20161174666350340422359855763, −3.19228863146850292738130299059, −2.68969660284431601718010562501, −2.29971433230300664828619821449, −1.93302415580359365720343952621, −1.78471396368394939442634467881, −1.75315506593520178628234586193, −1.15199731311521595090047502537, −0.871586509007980902735670380805, −0.60690585996871679307108533713, −0.27835296711710092997391852091,
0.27835296711710092997391852091, 0.60690585996871679307108533713, 0.871586509007980902735670380805, 1.15199731311521595090047502537, 1.75315506593520178628234586193, 1.78471396368394939442634467881, 1.93302415580359365720343952621, 2.29971433230300664828619821449, 2.68969660284431601718010562501, 3.19228863146850292738130299059, 3.20161174666350340422359855763, 3.21091273383519435551855237727, 3.56735951969467924482382384895, 3.59474957724721315819455328579, 4.23544502250961265704144209081, 4.26784528356735846969131880642, 4.57687490804278990761073451145, 4.86134871287296512603279605489, 4.93798733840619126917019149366, 4.97151904127886520704910668594, 5.51398904516286972249898819605, 5.64049901288981774754337544875, 5.78417763685452092199605828084, 5.80728262334941218987575966273, 6.25863523594975217737596065231
