L(s) = 1 | + 4·19-s + 7·25-s + 18·29-s + 2·31-s − 4·37-s − 18·53-s + 6·59-s + 30·83-s + 8·103-s − 8·109-s + 12·113-s − 5·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 22·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯ |
L(s) = 1 | + 0.917·19-s + 7/5·25-s + 3.34·29-s + 0.359·31-s − 0.657·37-s − 2.47·53-s + 0.781·59-s + 3.29·83-s + 0.788·103-s − 0.766·109-s + 1.12·113-s − 0.454·121-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 − 1.69·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 49787136 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49787136 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.195310647\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.195310647\) |
\(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
good | 5 | $C_2^2$ | \( 1 - 7 T^{2} + p^{2} T^{4} \) |
| 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 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 74 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 94 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 98 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 155 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 15 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 119 T^{2} + p^{2} T^{4} \) |
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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
−8.067930616678768791900548641803, −7.989810411622879333678554912670, −7.26148908030193934091228725329, −7.10701528032058803351037976610, −6.59685929290812151605973067048, −6.55875522052006382165980093148, −6.03648817959747602912665170080, −5.74529575725605960638861051570, −5.03851035977530608150476583489, −4.99512939636032358741777944117, −4.56460438780186228698963447386, −4.40757012903608050945101724566, −3.52843159595861769046141917367, −3.41048046352326861581144649524, −2.83348519623543581566665304856, −2.75509607995356418647957693714, −1.99636506646751035290470775935, −1.55332296604414735746812454651, −0.77993616409482370145314050783, −0.72185366516679199824611068990,
0.72185366516679199824611068990, 0.77993616409482370145314050783, 1.55332296604414735746812454651, 1.99636506646751035290470775935, 2.75509607995356418647957693714, 2.83348519623543581566665304856, 3.41048046352326861581144649524, 3.52843159595861769046141917367, 4.40757012903608050945101724566, 4.56460438780186228698963447386, 4.99512939636032358741777944117, 5.03851035977530608150476583489, 5.74529575725605960638861051570, 6.03648817959747602912665170080, 6.55875522052006382165980093148, 6.59685929290812151605973067048, 7.10701528032058803351037976610, 7.26148908030193934091228725329, 7.989810411622879333678554912670, 8.067930616678768791900548641803