| L(s) = 1 | + 6·5-s + 24·13-s + 16·17-s + 11·25-s + 42·29-s − 24·37-s − 80·41-s + 49·49-s + 90·53-s − 120·61-s + 144·65-s + 110·73-s + 96·85-s − 160·89-s + 130·97-s − 198·101-s + 120·109-s + 224·113-s + ⋯ |
| L(s) = 1 | + 6/5·5-s + 1.84·13-s + 0.941·17-s + 0.439·25-s + 1.44·29-s − 0.648·37-s − 1.95·41-s + 49-s + 1.69·53-s − 1.96·61-s + 2.21·65-s + 1.50·73-s + 1.12·85-s − 1.79·89-s + 1.34·97-s − 1.96·101-s + 1.10·109-s + 1.98·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(3.190175665\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.190175665\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 6 T + p^{2} T^{2} \) |
| 7 | \( ( 1 - p T )( 1 + p T ) \) |
| 11 | \( ( 1 - p T )( 1 + p T ) \) |
| 13 | \( 1 - 24 T + p^{2} T^{2} \) |
| 17 | \( 1 - 16 T + p^{2} T^{2} \) |
| 19 | \( ( 1 - p T )( 1 + p T ) \) |
| 23 | \( ( 1 - p T )( 1 + p T ) \) |
| 29 | \( 1 - 42 T + p^{2} T^{2} \) |
| 31 | \( ( 1 - p T )( 1 + p T ) \) |
| 37 | \( 1 + 24 T + p^{2} T^{2} \) |
| 41 | \( 1 + 80 T + p^{2} T^{2} \) |
| 43 | \( ( 1 - p T )( 1 + p T ) \) |
| 47 | \( ( 1 - p T )( 1 + p T ) \) |
| 53 | \( 1 - 90 T + p^{2} T^{2} \) |
| 59 | \( ( 1 - p T )( 1 + p T ) \) |
| 61 | \( 1 + 120 T + p^{2} T^{2} \) |
| 67 | \( ( 1 - p T )( 1 + p T ) \) |
| 71 | \( ( 1 - p T )( 1 + p T ) \) |
| 73 | \( 1 - 110 T + p^{2} T^{2} \) |
| 79 | \( ( 1 - p T )( 1 + p T ) \) |
| 83 | \( ( 1 - p T )( 1 + p T ) \) |
| 89 | \( 1 + 160 T + p^{2} T^{2} \) |
| 97 | \( 1 - 130 T + p^{2} T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.710634276745108101314063553844, −8.339516706782924632100821308831, −7.15300460810904002425620823161, −6.31701510267511286049281675433, −5.80428253650732973555339471293, −5.03143355652555553820922182364, −3.84724451033666849535441008850, −3.01000206916712094595855333218, −1.82053919368560567632576370298, −1.00719340357247899756831210829,
1.00719340357247899756831210829, 1.82053919368560567632576370298, 3.01000206916712094595855333218, 3.84724451033666849535441008850, 5.03143355652555553820922182364, 5.80428253650732973555339471293, 6.31701510267511286049281675433, 7.15300460810904002425620823161, 8.339516706782924632100821308831, 8.710634276745108101314063553844
