| L(s) = 1 | − 4-s + 6·11-s + 16-s − 16·19-s − 18·29-s − 14·31-s + 24·41-s − 6·44-s + 10·49-s + 24·59-s − 20·61-s − 64-s + 16·76-s + 2·79-s + 6·101-s + 8·109-s + 18·116-s + 5·121-s + 14·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + ⋯ |
| L(s) = 1 | − 1/2·4-s + 1.80·11-s + 1/4·16-s − 3.67·19-s − 3.34·29-s − 2.51·31-s + 3.74·41-s − 0.904·44-s + 10/7·49-s + 3.12·59-s − 2.56·61-s − 1/8·64-s + 1.83·76-s + 0.225·79-s + 0.597·101-s + 0.766·109-s + 1.67·116-s + 5/11·121-s + 1.25·124-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1822500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1822500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.115500875\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.115500875\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 37 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 37 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 85 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 118 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 142 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 158 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
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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
−9.528836584016703737346455993117, −9.195064991936558706921271508163, −9.162206934372555414531407627194, −8.744696125634607357875571074333, −8.393603165146125431052897436767, −7.63829359363180169247182684446, −7.40423700340470996439631574894, −7.06454382382721044355332746286, −6.27524251874756979416844504746, −6.24443137161075623110643982897, −5.65849471726723585656711300019, −5.38945733415161825648354142243, −4.35474864808621341791719021299, −4.25802655103044142804765450261, −3.75123337829033212102107866902, −3.72556206553507140747837691393, −2.43381526925528841979085487457, −2.08301401841255899416594754487, −1.55522591252549319444642721434, −0.43275255825030110116962641445,
0.43275255825030110116962641445, 1.55522591252549319444642721434, 2.08301401841255899416594754487, 2.43381526925528841979085487457, 3.72556206553507140747837691393, 3.75123337829033212102107866902, 4.25802655103044142804765450261, 4.35474864808621341791719021299, 5.38945733415161825648354142243, 5.65849471726723585656711300019, 6.24443137161075623110643982897, 6.27524251874756979416844504746, 7.06454382382721044355332746286, 7.40423700340470996439631574894, 7.63829359363180169247182684446, 8.393603165146125431052897436767, 8.744696125634607357875571074333, 9.162206934372555414531407627194, 9.195064991936558706921271508163, 9.528836584016703737346455993117
