| L(s) = 1 | + 4·13-s − 24·17-s − 24·29-s + 20·37-s + 24·49-s + 24·53-s + 20·61-s − 48·97-s + 24·101-s − 52·109-s − 24·113-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 8·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + ⋯ |
| L(s) = 1 | + 1.10·13-s − 5.82·17-s − 4.45·29-s + 3.28·37-s + 24/7·49-s + 3.29·53-s + 2.56·61-s − 4.87·97-s + 2.38·101-s − 4.98·109-s − 2.25·113-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 + 8/13·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.356807085\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.356807085\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 - 12 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2}( 1 + 4 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{4} \) |
| 19 | $C_2^3$ | \( 1 - 718 T^{4} + p^{4} T^{8} \) |
| 23 | $C_2^2$ | \( ( 1 + 26 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 + 12 T + 72 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 + 60 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2}( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 - 46 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^3$ | \( 1 - 1198 T^{4} + p^{4} T^{8} \) |
| 47 | $C_2^2$ | \( ( 1 + 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 12 T + 72 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 - 10 T + 50 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^3$ | \( 1 + 4946 T^{4} + p^{4} T^{8} \) |
| 71 | $C_2^2$ | \( ( 1 - 70 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 + 156 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^3$ | \( 1 - 13294 T^{4} + p^{4} T^{8} \) |
| 89 | $C_2^2$ | \( ( 1 - 142 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{4} \) |
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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
−6.46714280787397210439488723293, −6.17597995783632056657631792267, −5.92387775461146710296085899950, −5.71073416294572733335386709075, −5.70510918743473357131182352308, −5.34444544885517623226941439621, −5.32788029861272068291801847017, −4.87824814998532084020988560838, −4.61527511519245317811381811738, −4.27062819085745809150463364375, −4.09474367113339016159749577226, −4.07282440069341845717894324556, −4.06929763773270980659491834911, −3.70027810948321612485881886747, −3.64021752627419022289157079417, −2.87061068887513285319253244110, −2.67615584958346066760209047680, −2.45647611717807422851071426984, −2.29107723459708089329175967772, −2.12678682991458099044248908816, −2.03104027374006195044833112275, −1.35511617079620690035520217638, −1.20159309709098102279055442895, −0.52025880332249821125636964990, −0.27587155425655764035961868494,
0.27587155425655764035961868494, 0.52025880332249821125636964990, 1.20159309709098102279055442895, 1.35511617079620690035520217638, 2.03104027374006195044833112275, 2.12678682991458099044248908816, 2.29107723459708089329175967772, 2.45647611717807422851071426984, 2.67615584958346066760209047680, 2.87061068887513285319253244110, 3.64021752627419022289157079417, 3.70027810948321612485881886747, 4.06929763773270980659491834911, 4.07282440069341845717894324556, 4.09474367113339016159749577226, 4.27062819085745809150463364375, 4.61527511519245317811381811738, 4.87824814998532084020988560838, 5.32788029861272068291801847017, 5.34444544885517623226941439621, 5.70510918743473357131182352308, 5.71073416294572733335386709075, 5.92387775461146710296085899950, 6.17597995783632056657631792267, 6.46714280787397210439488723293
