| L(s) = 1 | + 32·11-s + 48·17-s + 48·19-s + 34·25-s − 16·41-s − 112·43-s + 82·49-s − 64·59-s + 160·67-s + 132·73-s + 32·83-s + 288·89-s + 188·97-s − 192·107-s − 160·113-s + 526·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 334·169-s + 173-s + ⋯ |
| L(s) = 1 | + 2.90·11-s + 2.82·17-s + 2.52·19-s + 1.35·25-s − 0.390·41-s − 2.60·43-s + 1.67·49-s − 1.08·59-s + 2.38·67-s + 1.80·73-s + 0.385·83-s + 3.23·89-s + 1.93·97-s − 1.79·107-s − 1.41·113-s + 4.34·121-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 + 1.97·169-s + 0.00578·173-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5308416 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5308416 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(7.117160037\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.117160037\) |
| \(L(2)\) |
|
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 | $C_2^2$ | \( 1 - 34 T^{2} + p^{4} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 82 T^{2} + p^{4} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 16 T + p^{2} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 334 T^{2} + p^{4} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 24 T + p^{2} T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 24 T + p^{2} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 34 T^{2} + p^{4} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 254 T^{2} + p^{4} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 782 T^{2} + p^{4} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 2414 T^{2} + p^{4} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 8 T + p^{2} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 56 T + p^{2} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 3394 T^{2} + p^{4} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 4322 T^{2} + p^{4} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 32 T + p^{2} T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 - 3598 T^{2} + p^{4} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 80 T + p^{2} T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 + 6302 T^{2} + p^{4} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 66 T + p^{2} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 12082 T^{2} + p^{4} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 16 T + p^{2} T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 144 T + p^{2} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 94 T + p^{2} T^{2} )^{2} \) |
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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
−9.124322221787193977733947345040, −8.796423283268401838081068333994, −8.052961487360610165113244991215, −7.987700107525749900502164227330, −7.43790308505832518744069314088, −7.09497041987748762785974135538, −6.55815077789208063616499392369, −6.52793485505650888076554146252, −5.89786985212802555939092851598, −5.38109454891458027114239568528, −5.07927290881580176477493525244, −4.84076913163196230346250078396, −3.79356479794869212902183790070, −3.77737415705294961830238600849, −3.29029575707643866158306587495, −3.10744133295739551758297754707, −2.11102590005701817201010711479, −1.38568041776269250114848438801, −0.986448082004347641098837275286, −0.919861306951436878314266719037,
0.919861306951436878314266719037, 0.986448082004347641098837275286, 1.38568041776269250114848438801, 2.11102590005701817201010711479, 3.10744133295739551758297754707, 3.29029575707643866158306587495, 3.77737415705294961830238600849, 3.79356479794869212902183790070, 4.84076913163196230346250078396, 5.07927290881580176477493525244, 5.38109454891458027114239568528, 5.89786985212802555939092851598, 6.52793485505650888076554146252, 6.55815077789208063616499392369, 7.09497041987748762785974135538, 7.43790308505832518744069314088, 7.987700107525749900502164227330, 8.052961487360610165113244991215, 8.796423283268401838081068333994, 9.124322221787193977733947345040
