| L(s) = 1 | + 2·2-s + 3·4-s + 4·8-s − 2·11-s + 5·16-s − 4·22-s − 2·25-s − 16·29-s + 6·32-s + 12·37-s + 8·43-s − 6·44-s − 4·50-s + 12·53-s − 32·58-s + 7·64-s − 8·67-s + 12·71-s + 24·74-s + 24·79-s + 16·86-s − 8·88-s − 6·100-s + 24·106-s − 12·107-s − 32·109-s − 28·113-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 3/2·4-s + 1.41·8-s − 0.603·11-s + 5/4·16-s − 0.852·22-s − 2/5·25-s − 2.97·29-s + 1.06·32-s + 1.97·37-s + 1.21·43-s − 0.904·44-s − 0.565·50-s + 1.64·53-s − 4.20·58-s + 7/8·64-s − 0.977·67-s + 1.42·71-s + 2.78·74-s + 2.70·79-s + 1.72·86-s − 0.852·88-s − 3/5·100-s + 2.33·106-s − 1.16·107-s − 3.06·109-s − 2.63·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 94128804 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 94128804 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.308189264\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.308189264\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 3 | | \( 1 \) |
| 7 | | \( 1 \) |
| 11 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 12 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 60 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 44 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 110 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 40 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 74 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 116 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 64 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 144 T^{2} + p^{2} T^{4} \) |
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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
−7.66581865878770086169520803952, −7.63362238073388942311955634834, −6.95409904947504520728113796786, −6.85653956103152510656696446210, −6.42388691104461960719565088928, −5.98614230469501452683895281319, −5.66037853163319986170706656983, −5.47713587041485881194559351213, −5.21461651676620261229250353823, −4.77714695405700510053804500914, −4.15724220995996450773411233157, −4.08799813158337190460095803665, −3.78623316392103469254275447438, −3.35297812324539984375493290858, −2.77528852017348765490566550807, −2.51814241954628856815631525932, −2.15208711338571205856404965829, −1.71261883885942142666576884402, −1.09588561555603425131919303949, −0.43077324012781284684716596598,
0.43077324012781284684716596598, 1.09588561555603425131919303949, 1.71261883885942142666576884402, 2.15208711338571205856404965829, 2.51814241954628856815631525932, 2.77528852017348765490566550807, 3.35297812324539984375493290858, 3.78623316392103469254275447438, 4.08799813158337190460095803665, 4.15724220995996450773411233157, 4.77714695405700510053804500914, 5.21461651676620261229250353823, 5.47713587041485881194559351213, 5.66037853163319986170706656983, 5.98614230469501452683895281319, 6.42388691104461960719565088928, 6.85653956103152510656696446210, 6.95409904947504520728113796786, 7.63362238073388942311955634834, 7.66581865878770086169520803952
