L(s) = 1 | + 8·19-s − 10·25-s − 8·31-s + 2·49-s − 28·61-s − 32·67-s + 20·73-s + 8·79-s − 40·103-s + 22·121-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 + 197-s + 199-s + ⋯ |
L(s) = 1 | + 1.83·19-s − 2·25-s − 1.43·31-s + 2/7·49-s − 3.58·61-s − 3.90·67-s + 2.34·73-s + 0.900·79-s − 3.94·103-s + 2·121-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 + 0.0712·197-s + 0.0708·199-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
\(L(1)\) |
\(\approx\) |
\(1.353172910\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.353172910\) |
\(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
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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
−9.258325169173506058864029629449, −8.535455777858778037774105948695, −8.315596968082283377717599883279, −7.59788129209117794402763414394, −7.59247474500704955408604414765, −7.35457138972783893423692290072, −6.80094788714854097119835323625, −6.09806107116109456996088568771, −6.05477287361117004818924226404, −5.50950867430770717917912748299, −5.29135597604977596263929489311, −4.52195966249771805138893023668, −4.47042951013442703450900285351, −3.63030653343448141276192639751, −3.48251934844891486970631108746, −2.95207365975957457267832019900, −2.40050699229020987835106063028, −1.62946041223643571654406370624, −1.45485945549984179192229020817, −0.37413348455242955654809458644,
0.37413348455242955654809458644, 1.45485945549984179192229020817, 1.62946041223643571654406370624, 2.40050699229020987835106063028, 2.95207365975957457267832019900, 3.48251934844891486970631108746, 3.63030653343448141276192639751, 4.47042951013442703450900285351, 4.52195966249771805138893023668, 5.29135597604977596263929489311, 5.50950867430770717917912748299, 6.05477287361117004818924226404, 6.09806107116109456996088568771, 6.80094788714854097119835323625, 7.35457138972783893423692290072, 7.59247474500704955408604414765, 7.59788129209117794402763414394, 8.315596968082283377717599883279, 8.535455777858778037774105948695, 9.258325169173506058864029629449