| L(s) = 1 | + 4·9-s + 456·17-s − 50·25-s − 1.36e3·41-s − 1.26e3·49-s − 2.72e3·73-s − 1.44e3·81-s + 2.52e3·89-s − 3.86e3·97-s + 8.42e3·113-s + 1.58e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 1.82e3·153-s + 157-s + 163-s + 167-s + 6.47e3·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯ |
| L(s) = 1 | + 4/27·9-s + 6.50·17-s − 2/5·25-s − 5.21·41-s − 3.69·49-s − 4.37·73-s − 1.98·81-s + 3.00·89-s − 4.04·97-s + 7.01·113-s + 1.18·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + 0.963·153-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s + 2.94·169-s + 0.000439·173-s + 0.000417·179-s + 0.000410·181-s + 0.000378·191-s + 0.000372·193-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.04048820875\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.04048820875\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| good | 3 | $C_2^2$ | \( ( 1 - 2 T^{2} + p^{6} T^{4} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 + 634 T^{2} + p^{6} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 790 T^{2} + p^{6} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 3238 T^{2} + p^{6} T^{4} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 114 T + p^{3} T^{2} )^{4} \) |
| 19 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{4} \) |
| 23 | $C_2^2$ | \( ( 1 - 19398 T^{2} + p^{6} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 48102 T^{2} + p^{6} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 + 49390 T^{2} + p^{6} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 - 78806 T^{2} + p^{6} T^{4} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 342 T + p^{3} T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 + 47374 T^{2} + p^{6} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 - 133526 T^{2} + p^{6} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 229110 T^{2} + p^{6} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 170310 T^{2} + p^{6} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 - 385318 T^{2} + p^{6} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 - 353954 T^{2} + p^{6} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - 392610 T^{2} + p^{6} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 682 T + p^{3} T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 + 945310 T^{2} + p^{6} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 1120642 T^{2} + p^{6} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 630 T + p^{3} T^{2} )^{4} \) |
| 97 | $C_2$ | \( ( 1 + 966 T + p^{3} T^{2} )^{4} \) |
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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
−6.50258259341086767939708751275, −6.14170072445042231768789989819, −6.10713592879840236663268572507, −5.66209440992651868886927563326, −5.64620369546761666052061883320, −5.48589488246127626059505607420, −5.31020388688181141736455021869, −4.97988942885859018321232205913, −4.75187199868678485608668198746, −4.50721839403613089409437576624, −4.44355395656992058702262749089, −3.70901274842852779415522308150, −3.56276854004489452278071250208, −3.34820003041029563179613764990, −3.33654644397825171677479493265, −3.14833983324353832820653196127, −3.01511542500937931626679540163, −2.49468953586727456224238906467, −1.96646800525606521369640884382, −1.59463056005120232408201786769, −1.42890759140670638817856235779, −1.39497046939327664504885408973, −1.06438900628195476885884664336, −0.58533824511319404912454258055, −0.02148336313754888626262867966,
0.02148336313754888626262867966, 0.58533824511319404912454258055, 1.06438900628195476885884664336, 1.39497046939327664504885408973, 1.42890759140670638817856235779, 1.59463056005120232408201786769, 1.96646800525606521369640884382, 2.49468953586727456224238906467, 3.01511542500937931626679540163, 3.14833983324353832820653196127, 3.33654644397825171677479493265, 3.34820003041029563179613764990, 3.56276854004489452278071250208, 3.70901274842852779415522308150, 4.44355395656992058702262749089, 4.50721839403613089409437576624, 4.75187199868678485608668198746, 4.97988942885859018321232205913, 5.31020388688181141736455021869, 5.48589488246127626059505607420, 5.64620369546761666052061883320, 5.66209440992651868886927563326, 6.10713592879840236663268572507, 6.14170072445042231768789989819, 6.50258259341086767939708751275
