L(s) = 1 | + 2·9-s + 8·11-s − 8·23-s − 8·25-s + 16·29-s − 16·37-s − 8·43-s + 20·53-s + 16·79-s − 5·81-s + 16·99-s + 16·107-s − 16·109-s + 12·113-s + 26·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 8·169-s + 173-s + 179-s + ⋯ |
L(s) = 1 | + 2/3·9-s + 2.41·11-s − 1.66·23-s − 8/5·25-s + 2.97·29-s − 2.63·37-s − 1.21·43-s + 2.74·53-s + 1.80·79-s − 5/9·81-s + 1.60·99-s + 1.54·107-s − 1.53·109-s + 1.12·113-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 + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 0.615·169-s + 0.0760·173-s + 0.0747·179-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38416 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38416 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.508747085\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.508747085\) |
\(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | | \( 1 \) |
good | 3 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 32 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 32 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 72 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 96 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 128 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 192 T^{2} + p^{2} T^{4} \) |
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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
−12.50186006834687765954439285399, −12.01801209257579565799880886231, −11.78118085089026838605485024886, −11.76714388312700296463567101866, −10.58045995448057079474808428178, −10.24476681434007141548813779353, −9.877393439712082231162545028013, −9.329779081065567815132494777511, −8.587224786167342026200073187279, −8.487543707786345328675481589703, −7.65409378345593133485819794447, −6.89650934972302960219554185576, −6.61424528978563222744386823329, −6.14539973457519346514706405452, −5.33135340261650426661592219336, −4.50238113647116466674969041672, −3.91584139815081451345714498381, −3.55451294290711262068121905472, −2.16495590463024630762628072114, −1.32286566317686927886391412472,
1.32286566317686927886391412472, 2.16495590463024630762628072114, 3.55451294290711262068121905472, 3.91584139815081451345714498381, 4.50238113647116466674969041672, 5.33135340261650426661592219336, 6.14539973457519346514706405452, 6.61424528978563222744386823329, 6.89650934972302960219554185576, 7.65409378345593133485819794447, 8.487543707786345328675481589703, 8.587224786167342026200073187279, 9.329779081065567815132494777511, 9.877393439712082231162545028013, 10.24476681434007141548813779353, 10.58045995448057079474808428178, 11.76714388312700296463567101866, 11.78118085089026838605485024886, 12.01801209257579565799880886231, 12.50186006834687765954439285399