| L(s) = 1 | + 4·9-s − 12·17-s + 14·25-s + 24·41-s − 22·49-s − 4·73-s − 6·81-s + 16·97-s − 72·113-s + 26·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 48·153-s + 157-s + 163-s + 167-s + 52·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + ⋯ |
| L(s) = 1 | + 4/3·9-s − 2.91·17-s + 14/5·25-s + 3.74·41-s − 3.14·49-s − 0.468·73-s − 2/3·81-s + 1.62·97-s − 6.77·113-s + 2.36·121-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 − 3.88·153-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 4·169-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.749143263\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.749143263\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 19 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| good | 3 | $C_2^2$ | \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 5 | $C_2^2$ | \( ( 1 - 7 T^{2} + p^{2} T^{4} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 + 11 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 13 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 17 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{4} \) |
| 23 | $C_2^2$ | \( ( 1 + 34 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 46 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 + 34 T^{2} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 - 85 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + 67 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 59 | $C_2^2$ | \( ( 1 - 82 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 - 95 T^{2} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 - 118 T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 + 130 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + T + p T^{2} )^{4} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 97 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{4} \) |
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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
−6.90739691683867808223964668136, −6.67083769185318224221616767988, −6.54061604236891968259855100544, −6.50430011219516939125375699798, −6.20004686601810393127131810002, −5.97740398979658914658892958820, −5.52567955327388282498926208982, −5.30996495404645124766230089083, −5.13484712119125654347457659588, −4.91954337306519218008702268791, −4.57629126388276573532465852878, −4.36641153913946972863328227426, −4.26152648992979132034159970546, −4.04664179287802178979533292477, −4.01092049735730250287419843736, −3.30172783207716918971575777861, −3.05900881158751192129317369803, −2.79525138281833089959285908556, −2.78433399964219420287668859552, −2.14464037007539385288493675998, −2.10934568023208294799193817828, −1.58264610748522764387088759791, −1.34366561065017568858271144626, −0.869532506126091502286124386544, −0.38882321693785586860652856629,
0.38882321693785586860652856629, 0.869532506126091502286124386544, 1.34366561065017568858271144626, 1.58264610748522764387088759791, 2.10934568023208294799193817828, 2.14464037007539385288493675998, 2.78433399964219420287668859552, 2.79525138281833089959285908556, 3.05900881158751192129317369803, 3.30172783207716918971575777861, 4.01092049735730250287419843736, 4.04664179287802178979533292477, 4.26152648992979132034159970546, 4.36641153913946972863328227426, 4.57629126388276573532465852878, 4.91954337306519218008702268791, 5.13484712119125654347457659588, 5.30996495404645124766230089083, 5.52567955327388282498926208982, 5.97740398979658914658892958820, 6.20004686601810393127131810002, 6.50430011219516939125375699798, 6.54061604236891968259855100544, 6.67083769185318224221616767988, 6.90739691683867808223964668136
