| L(s) = 1 | − 16·23-s − 8·25-s + 48·47-s + 32·49-s + 16·71-s − 16·73-s + 24·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 40·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + ⋯ |
| L(s) = 1 | − 3.33·23-s − 8/5·25-s + 7.00·47-s + 32/7·49-s + 1.89·71-s − 1.87·73-s + 2.18·121-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 + 3.07·169-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 + 0.0708·199-s + 0.0688·211-s + 0.0669·223-s + 0.0663·227-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{72} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{72} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(17.34545092\) |
| \(L(\frac12)\) |
\(\approx\) |
\(17.34545092\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( ( 1 + 4 T^{2} + 22 T^{4} + 4 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 7 | \( ( 1 - 16 T^{2} + 130 T^{4} - 16 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 11 | \( ( 1 - 12 T^{2} + 150 T^{4} - 12 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 13 | \( ( 1 - 20 T^{2} + 310 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 17 | \( ( 1 - 32 T^{2} + p^{2} T^{4} )^{4} \) |
| 19 | \( ( 1 + 12 T^{2} + 246 T^{4} + 12 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 23 | \( ( 1 + 4 T + 42 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{4} \) |
| 29 | \( ( 1 + 36 T^{2} + 1974 T^{4} + 36 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 31 | \( ( 1 - 80 T^{2} + 3234 T^{4} - 80 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 37 | \( ( 1 - 84 T^{2} + 3990 T^{4} - 84 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 41 | \( ( 1 - 16 T + 128 T^{2} - 16 p T^{3} + p^{2} T^{4} )^{2}( 1 + 16 T + 128 T^{2} + 16 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | \( ( 1 + 44 T^{2} + 2134 T^{4} + 44 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 47 | \( ( 1 - 12 T + 122 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{4} \) |
| 53 | \( ( 1 + 132 T^{2} + 9942 T^{4} + 132 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 59 | \( ( 1 - 108 T^{2} + 7830 T^{4} - 108 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 - 180 T^{2} + 15030 T^{4} - 180 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 67 | \( ( 1 - 52 T^{2} + 9142 T^{4} - 52 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 71 | \( ( 1 - 4 T + 74 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{4} \) |
| 73 | \( ( 1 + 4 T + 118 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{4} \) |
| 79 | \( ( 1 - 16 T^{2} - 7454 T^{4} - 16 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 83 | \( ( 1 - 172 T^{2} + 21046 T^{4} - 172 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 89 | \( ( 1 - 128 T^{2} + p^{2} T^{4} )^{4} \) |
| 97 | \( ( 1 + 66 T^{2} + p^{2} T^{4} )^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.55426768716940718768672498480, −3.24341504206632328166575249844, −3.22777451455141064225712983876, −2.88646261209936271145893772541, −2.86777363395290743342732725076, −2.79955623023214714753017483720, −2.79313889536515809567635511860, −2.62659865361996172567169989327, −2.48740467597407381241526980390, −2.25022292147112616880873068738, −2.20488667342369414921644699153, −2.17614373710752862112015043416, −2.09068133914807457610018119613, −1.83815858313693995592456221291, −1.77462599007617442061477646059, −1.76276783993932204039590001637, −1.63125226806844520952276749837, −1.21709129352895411468549864824, −1.05746499287135226031843267910, −1.03555708594120248975021312132, −0.867626002596034630020483506329, −0.61718887125691216691961562076, −0.45873411784773335247424215848, −0.41429391973192471534910359749, −0.33436857845082718818804864441,
0.33436857845082718818804864441, 0.41429391973192471534910359749, 0.45873411784773335247424215848, 0.61718887125691216691961562076, 0.867626002596034630020483506329, 1.03555708594120248975021312132, 1.05746499287135226031843267910, 1.21709129352895411468549864824, 1.63125226806844520952276749837, 1.76276783993932204039590001637, 1.77462599007617442061477646059, 1.83815858313693995592456221291, 2.09068133914807457610018119613, 2.17614373710752862112015043416, 2.20488667342369414921644699153, 2.25022292147112616880873068738, 2.48740467597407381241526980390, 2.62659865361996172567169989327, 2.79313889536515809567635511860, 2.79955623023214714753017483720, 2.86777363395290743342732725076, 2.88646261209936271145893772541, 3.22777451455141064225712983876, 3.24341504206632328166575249844, 3.55426768716940718768672498480
Plot not available for L-functions of degree greater than 10.
