| L(s) = 1 | + 2·5-s − 2·7-s + 14·17-s + 2·25-s − 4·35-s + 4·37-s − 10·41-s + 12·43-s + 4·47-s − 3·49-s − 20·67-s + 4·79-s + 4·83-s + 28·85-s − 10·89-s + 2·101-s + 12·109-s − 28·119-s + 10·121-s + 10·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 0.755·7-s + 3.39·17-s + 2/5·25-s − 0.676·35-s + 0.657·37-s − 1.56·41-s + 1.82·43-s + 0.583·47-s − 3/7·49-s − 2.44·67-s + 0.450·79-s + 0.439·83-s + 3.03·85-s − 1.05·89-s + 0.199·101-s + 1.14·109-s − 2.56·119-s + 0.909·121-s + 0.894·125-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 254016 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 254016 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.138728369\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.138728369\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 + 8 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 97 | $C_2^2$ | \( 1 + 114 T^{2} + p^{2} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.159134956812869679568273723662, −8.463305659030048724007518206089, −7.920265203540263934984211441912, −7.51943033712456739689546521679, −7.14936699479032740754868333143, −6.36722283838272124419966171712, −6.00971961788740877356019616363, −5.59074900788876333624744729444, −5.24308953152480817322793849162, −4.47425799401230125786402841817, −3.69705392804969212916691785128, −3.17530297564066487490438891901, −2.75960732049153766581847437261, −1.71508071193189601758216486522, −0.959093063501284743139378785566,
0.959093063501284743139378785566, 1.71508071193189601758216486522, 2.75960732049153766581847437261, 3.17530297564066487490438891901, 3.69705392804969212916691785128, 4.47425799401230125786402841817, 5.24308953152480817322793849162, 5.59074900788876333624744729444, 6.00971961788740877356019616363, 6.36722283838272124419966171712, 7.14936699479032740754868333143, 7.51943033712456739689546521679, 7.920265203540263934984211441912, 8.463305659030048724007518206089, 9.159134956812869679568273723662
