| L(s) = 1 | − 2·16-s − 16·31-s + 112·61-s + 9·81-s − 20·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-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 + 239-s + ⋯ |
| L(s) = 1 | − 1/2·16-s − 2.87·31-s + 14.3·61-s + 81-s − 1.81·121-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 + 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 + 0.0646·239-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.409898044\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.409898044\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 + T^{4} )^{2} \) |
| 3 | \( 1 - p^{2} T^{4} + p^{4} T^{8} \) |
| 5 | \( 1 \) |
| good | 7 | \( ( 1 - 94 T^{4} + p^{4} T^{8} )^{2} \) |
| 11 | \( ( 1 + 5 T^{2} + p^{2} T^{4} )^{4} \) |
| 13 | \( ( 1 + p^{2} T^{4} )^{4} \) |
| 17 | \( ( 1 + 47 T^{4} + p^{4} T^{8} )^{2} \) |
| 19 | \( ( 1 - 37 T^{2} + p^{2} T^{4} )^{4} \) |
| 23 | \( ( 1 - 958 T^{4} + p^{4} T^{8} )^{2} \) |
| 29 | \( ( 1 + p T^{2} )^{8} \) |
| 31 | \( ( 1 + 2 T + p T^{2} )^{8} \) |
| 37 | \( ( 1 + 1106 T^{4} + p^{4} T^{8} )^{2} \) |
| 41 | \( ( 1 - 55 T^{2} + p^{2} T^{4} )^{4} \) |
| 43 | \( ( 1 + 1778 T^{4} + p^{4} T^{8} )^{2} \) |
| 47 | \( ( 1 + p^{2} T^{4} )^{4} \) |
| 53 | \( ( 1 - 718 T^{4} + p^{4} T^{8} )^{2} \) |
| 59 | \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{4} \) |
| 61 | \( ( 1 - 14 T + p T^{2} )^{8} \) |
| 67 | \( ( 1 + 2471 T^{4} + p^{4} T^{8} )^{2} \) |
| 71 | \( ( 1 - p T^{2} )^{8} \) |
| 73 | \( ( 1 - 5617 T^{4} + p^{4} T^{8} )^{2} \) |
| 79 | \( ( 1 + 38 T^{2} + p^{2} T^{4} )^{4} \) |
| 83 | \( ( 1 + 10871 T^{4} + p^{4} T^{8} )^{2} \) |
| 89 | \( ( 1 - 65 T^{2} + p^{2} T^{4} )^{4} \) |
| 97 | \( ( 1 + 2498 T^{4} + p^{4} T^{8} )^{2} \) |
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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
−6.05292294595787903370532722539, −5.65662929676451807619200184444, −5.51721419258324170294415894852, −5.50210182948814418269338715932, −5.46264046827006235950537540646, −5.29478349274904067171114416408, −5.14095023665045280523795529715, −4.94559894868430505510387522605, −4.80060521669202178668491220590, −4.59037963425345594477193943323, −4.08571258127475767880558726187, −4.00531687195082858702010787900, −3.87838820991488060311248225288, −3.75160830104294674073228019702, −3.74289328770649006938144316249, −3.61963447891937566463349296429, −3.23893213817240536143298662573, −2.70572854985074972695335552668, −2.62999409128695463453044170967, −2.47033492955680976088709044635, −2.21477996751888029364567173423, −1.96713013012517432355421949032, −1.82213642098685371348147253708, −1.05289566278120391432237533337, −0.77580983440891378740683602000,
0.77580983440891378740683602000, 1.05289566278120391432237533337, 1.82213642098685371348147253708, 1.96713013012517432355421949032, 2.21477996751888029364567173423, 2.47033492955680976088709044635, 2.62999409128695463453044170967, 2.70572854985074972695335552668, 3.23893213817240536143298662573, 3.61963447891937566463349296429, 3.74289328770649006938144316249, 3.75160830104294674073228019702, 3.87838820991488060311248225288, 4.00531687195082858702010787900, 4.08571258127475767880558726187, 4.59037963425345594477193943323, 4.80060521669202178668491220590, 4.94559894868430505510387522605, 5.14095023665045280523795529715, 5.29478349274904067171114416408, 5.46264046827006235950537540646, 5.50210182948814418269338715932, 5.51721419258324170294415894852, 5.65662929676451807619200184444, 6.05292294595787903370532722539
Plot not available for L-functions of degree greater than 10.
