L(s) = 1 | − 2·3-s + 3·9-s − 8·25-s − 4·27-s + 8·31-s − 8·37-s + 24·47-s − 20·53-s + 16·75-s + 5·81-s − 8·83-s − 16·93-s + 24·103-s + 8·109-s + 16·111-s − 4·113-s − 14·121-s + 127-s + 131-s + 137-s + 139-s − 48·141-s + 149-s + 151-s + 157-s + 40·159-s + 163-s + ⋯ |
L(s) = 1 | − 1.15·3-s + 9-s − 8/5·25-s − 0.769·27-s + 1.43·31-s − 1.31·37-s + 3.50·47-s − 2.74·53-s + 1.84·75-s + 5/9·81-s − 0.878·83-s − 1.65·93-s + 2.36·103-s + 0.766·109-s + 1.51·111-s − 0.376·113-s − 1.27·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 4.04·141-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 3.17·159-s + 0.0783·163-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 88510464 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 88510464 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.399656483\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.399656483\) |
\(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 7 | | \( 1 \) |
good | 5 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 24 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 32 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 80 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 54 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 + 120 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 134 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 16 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 128 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 96 T^{2} + p^{2} T^{4} \) |
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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
−7.65073084407784411736313931587, −7.51459575424776855093613593925, −7.23429679082443462535994273216, −6.79233857099951129486591792206, −6.37135339811430415999230939353, −6.20250335783729617003269280733, −5.74342477355179153731959925307, −5.66781468334466374615626317060, −5.10647592199974558494119292523, −4.82554840487620804777992537324, −4.39048783354774690280236127702, −4.20246274677335008777852687988, −3.56701637997582601146434517321, −3.49106880330297909021481093433, −2.63379032368089645045389597907, −2.51901954319678542874303631599, −1.68751951739619551327083149121, −1.59364298498603317639862260818, −0.78096710584674172255292214303, −0.38356289442500274545899260156,
0.38356289442500274545899260156, 0.78096710584674172255292214303, 1.59364298498603317639862260818, 1.68751951739619551327083149121, 2.51901954319678542874303631599, 2.63379032368089645045389597907, 3.49106880330297909021481093433, 3.56701637997582601146434517321, 4.20246274677335008777852687988, 4.39048783354774690280236127702, 4.82554840487620804777992537324, 5.10647592199974558494119292523, 5.66781468334466374615626317060, 5.74342477355179153731959925307, 6.20250335783729617003269280733, 6.37135339811430415999230939353, 6.79233857099951129486591792206, 7.23429679082443462535994273216, 7.51459575424776855093613593925, 7.65073084407784411736313931587