| L(s) = 1 | + 32·25-s − 16·37-s − 18·81-s + 16·109-s − 40·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 32·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + ⋯ |
| L(s) = 1 | + 32/5·25-s − 2.63·37-s − 2·81-s + 1.53·109-s − 3.63·121-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 − 2.46·169-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 + 0.0660·229-s + 0.0655·233-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{8} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{8} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6629735724\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6629735724\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 7 | \( 1 \) |
| good | 5 | \( ( 1 - 8 T^{2} + p^{2} T^{4} )^{4} \) |
| 11 | \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{4} \) |
| 13 | \( ( 1 + 8 T^{2} + p^{2} T^{4} )^{4} \) |
| 17 | \( ( 1 - 26 T^{2} + p^{2} T^{4} )^{4} \) |
| 19 | \( ( 1 - 32 T^{2} + p^{2} T^{4} )^{4} \) |
| 23 | \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{4} \) |
| 29 | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{4} \) |
| 31 | \( ( 1 - 10 T + p T^{2} )^{4}( 1 + 10 T + p T^{2} )^{4} \) |
| 37 | \( ( 1 + 2 T + p T^{2} )^{8} \) |
| 41 | \( ( 1 - 6 T + p T^{2} )^{4}( 1 + 6 T + p T^{2} )^{4} \) |
| 43 | \( ( 1 - 74 T^{2} + p^{2} T^{4} )^{4} \) |
| 47 | \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{4} \) |
| 53 | \( ( 1 - 70 T^{2} + p^{2} T^{4} )^{4} \) |
| 59 | \( ( 1 + 112 T^{2} + p^{2} T^{4} )^{4} \) |
| 61 | \( ( 1 - 40 T^{2} + p^{2} T^{4} )^{4} \) |
| 67 | \( ( 1 - 26 T^{2} + p^{2} T^{4} )^{4} \) |
| 71 | \( ( 1 + 130 T^{2} + p^{2} T^{4} )^{4} \) |
| 73 | \( ( 1 + 74 T^{2} + p^{2} T^{4} )^{4} \) |
| 79 | \( ( 1 - 110 T^{2} + p^{2} T^{4} )^{4} \) |
| 83 | \( ( 1 + 16 T^{2} + p^{2} T^{4} )^{4} \) |
| 89 | \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{4} \) |
| 97 | \( ( 1 - 94 T^{2} + p^{2} T^{4} )^{4} \) |
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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
−3.74661559581497975464513429396, −3.53678599124572181730712034483, −3.51144985885332186923987104633, −3.41192938693996993029903819967, −3.18231154647436053273380177405, −3.09475936499154288696031213466, −3.02321535167228978990256213438, −2.77501800568246459105655381768, −2.71348790898077062829053216392, −2.70475655136478116611356276956, −2.69545957376333352088273917834, −2.66001645884032694035622617840, −2.21550492714962441161005182348, −1.98733007613650670974653497305, −1.91654593720374808337536982396, −1.75457436805935884087324180433, −1.65913484160011450569361358311, −1.60365201317002722468835428797, −1.33612549510442285680387125855, −1.08399929940522489411204473545, −1.01990818015335624369087166736, −0.851764202729373369112778698645, −0.60383625505249351877384341786, −0.56583470730083428050492873817, −0.05660829959807834371137121516,
0.05660829959807834371137121516, 0.56583470730083428050492873817, 0.60383625505249351877384341786, 0.851764202729373369112778698645, 1.01990818015335624369087166736, 1.08399929940522489411204473545, 1.33612549510442285680387125855, 1.60365201317002722468835428797, 1.65913484160011450569361358311, 1.75457436805935884087324180433, 1.91654593720374808337536982396, 1.98733007613650670974653497305, 2.21550492714962441161005182348, 2.66001645884032694035622617840, 2.69545957376333352088273917834, 2.70475655136478116611356276956, 2.71348790898077062829053216392, 2.77501800568246459105655381768, 3.02321535167228978990256213438, 3.09475936499154288696031213466, 3.18231154647436053273380177405, 3.41192938693996993029903819967, 3.51144985885332186923987104633, 3.53678599124572181730712034483, 3.74661559581497975464513429396
Plot not available for L-functions of degree greater than 10.
