| L(s) = 1 | − 6·5-s − 6·17-s + 8·19-s + 17·25-s − 8·31-s + 11·49-s − 24·59-s + 14·61-s + 16·67-s − 24·71-s − 10·73-s − 16·79-s + 36·85-s − 48·95-s − 12·101-s − 16·103-s − 5·121-s − 18·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 48·155-s + 157-s + 163-s + ⋯ |
| L(s) = 1 | − 2.68·5-s − 1.45·17-s + 1.83·19-s + 17/5·25-s − 1.43·31-s + 11/7·49-s − 3.12·59-s + 1.79·61-s + 1.95·67-s − 2.84·71-s − 1.17·73-s − 1.80·79-s + 3.90·85-s − 4.92·95-s − 1.19·101-s − 1.57·103-s − 0.454·121-s − 1.60·125-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 + 3.85·155-s + 0.0798·157-s + 0.0783·163-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7485696 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7485696 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1628115080\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1628115080\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 19 | $C_2$ | \( 1 - 8 T + p T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 + 5 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 11 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 19 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 154 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 146 T^{2} + p^{2} T^{4} \) |
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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
−9.212291945496614031425153989737, −8.558571977932800597808796766576, −8.288286964029819327067385635200, −7.64507147202340724519181813769, −7.60665540134120069815031270274, −7.20448437257015446406325909788, −7.05976695935073563409145620430, −6.46924621122733409412709229248, −5.93724806769214772529088030741, −5.38686810365043352245533050439, −5.13698122377714035576864543291, −4.41426893708808648998914503636, −4.24188192936261500795265067956, −3.93691725114823921101828932161, −3.44885171456428312088057428451, −3.01558305608033511446890935733, −2.62321827240139958344867851899, −1.72354329503954401126436974481, −1.05701477312683819753789372682, −0.15325068011453699940967475640,
0.15325068011453699940967475640, 1.05701477312683819753789372682, 1.72354329503954401126436974481, 2.62321827240139958344867851899, 3.01558305608033511446890935733, 3.44885171456428312088057428451, 3.93691725114823921101828932161, 4.24188192936261500795265067956, 4.41426893708808648998914503636, 5.13698122377714035576864543291, 5.38686810365043352245533050439, 5.93724806769214772529088030741, 6.46924621122733409412709229248, 7.05976695935073563409145620430, 7.20448437257015446406325909788, 7.60665540134120069815031270274, 7.64507147202340724519181813769, 8.288286964029819327067385635200, 8.558571977932800597808796766576, 9.212291945496614031425153989737
