| L(s) = 1 | − 2·4-s − 2·9-s + 3·16-s − 16·19-s − 24·29-s + 16·31-s + 4·36-s + 8·41-s − 20·49-s + 8·61-s − 4·64-s − 16·71-s + 32·76-s − 16·79-s + 3·81-s − 8·89-s − 56·101-s − 8·109-s + 48·116-s − 44·121-s − 32·124-s + 127-s + 131-s + 137-s + 139-s − 6·144-s + 149-s + ⋯ |
| L(s) = 1 | − 4-s − 2/3·9-s + 3/4·16-s − 3.67·19-s − 4.45·29-s + 2.87·31-s + 2/3·36-s + 1.24·41-s − 2.85·49-s + 1.02·61-s − 1/2·64-s − 1.89·71-s + 3.67·76-s − 1.80·79-s + 1/3·81-s − 0.847·89-s − 5.57·101-s − 0.766·109-s + 4.45·116-s − 4·121-s − 2.87·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 1/2·144-s + 0.0819·149-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{8} \cdot 17^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{8} \cdot 17^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2801826494\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2801826494\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 3 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 5 | | \( 1 \) |
| 17 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| good | 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2}( 1 + 2 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 13 | $D_4\times C_2$ | \( 1 + 4 T^{2} - 42 T^{4} + 4 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) |
| 23 | $C_2^2$ | \( ( 1 - 30 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{4} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{4} \) |
| 37 | $C_2^2$ | \( ( 1 - 38 T^{2} + p^{2} T^{4} )^{2} \) |
| 41 | $D_{4}$ | \( ( 1 - 4 T + 62 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 92 T^{2} + 4278 T^{4} - 92 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $C_2^2$ | \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $D_4\times C_2$ | \( 1 - 12 T^{2} + 4118 T^{4} - 12 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $C_2^2$ | \( ( 1 + 94 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 - 4 T + 30 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 188 T^{2} + 16278 T^{4} - 188 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 + 8 T + 134 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 172 T^{2} + 14598 T^{4} - 172 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 + 8 T + 78 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 108 T^{2} + 10550 T^{4} - 108 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 + 4 T + 86 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 + 52 T^{2} + 16038 T^{4} + 52 p^{2} T^{6} + p^{4} T^{8} \) |
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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
−6.36011126016323038697867435359, −5.96059103336306500130932082861, −5.87987691284628707246006574945, −5.71308281736953227703399701180, −5.62657916734905469164180277613, −5.25227805831730867440728803842, −5.08516545346670090866997663480, −4.83625050402466684097425728755, −4.46726294443081497500882844724, −4.42950169149649833523900349573, −4.25690944728786565723072602545, −4.01559655696733607028704117303, −3.85383124520075849115852723334, −3.59533071939323394580718895463, −3.46063575178426086665261378584, −2.84309354136588523945401882728, −2.80885593434393858770740299818, −2.52881151657490321322747068675, −2.44875501827474828794991139577, −1.83710965086485741568466836473, −1.80624680830123940465618460201, −1.29622532570298227125708074685, −1.29040103228238986982684350441, −0.26362376804872459902107068556, −0.23029475477837846189140319816,
0.23029475477837846189140319816, 0.26362376804872459902107068556, 1.29040103228238986982684350441, 1.29622532570298227125708074685, 1.80624680830123940465618460201, 1.83710965086485741568466836473, 2.44875501827474828794991139577, 2.52881151657490321322747068675, 2.80885593434393858770740299818, 2.84309354136588523945401882728, 3.46063575178426086665261378584, 3.59533071939323394580718895463, 3.85383124520075849115852723334, 4.01559655696733607028704117303, 4.25690944728786565723072602545, 4.42950169149649833523900349573, 4.46726294443081497500882844724, 4.83625050402466684097425728755, 5.08516545346670090866997663480, 5.25227805831730867440728803842, 5.62657916734905469164180277613, 5.71308281736953227703399701180, 5.87987691284628707246006574945, 5.96059103336306500130932082861, 6.36011126016323038697867435359
