| L(s) = 1 | + 2·2-s − 2·3-s + 3·4-s − 4·6-s − 2·7-s + 4·8-s + 2·11-s − 6·12-s − 4·13-s − 4·14-s + 5·16-s − 2·19-s + 4·21-s + 4·22-s − 6·23-s − 8·24-s − 8·26-s + 2·27-s − 6·28-s − 6·29-s + 4·31-s + 6·32-s − 4·33-s − 10·37-s − 4·38-s + 8·39-s − 6·41-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 1.15·3-s + 3/2·4-s − 1.63·6-s − 0.755·7-s + 1.41·8-s + 0.603·11-s − 1.73·12-s − 1.10·13-s − 1.06·14-s + 5/4·16-s − 0.458·19-s + 0.872·21-s + 0.852·22-s − 1.25·23-s − 1.63·24-s − 1.56·26-s + 0.384·27-s − 1.13·28-s − 1.11·29-s + 0.718·31-s + 1.06·32-s − 0.696·33-s − 1.64·37-s − 0.648·38-s + 1.28·39-s − 0.937·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14822500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14822500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 5 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 + T )^{2} \) |
| 11 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 + 2 T + 4 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 22 T^{2} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 2 T + 36 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 6 T + 52 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 6 T + 64 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 10 T + 96 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 6 T + 88 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 4 T + 42 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $D_{4}$ | \( 1 - 6 T + 112 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 74 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 67 | $D_{4}$ | \( 1 - 8 T + 102 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 12 T + 166 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 16 T + 198 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 10 T + 180 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 12 T + 94 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 24 T + 310 T^{2} + 24 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 22 T + 288 T^{2} + 22 p T^{3} + p^{2} T^{4} \) |
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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
−8.066233460339454264682989348086, −7.939959566335257094233292334813, −7.27620223359216700117805352198, −6.85168186746843464535776811980, −6.69946625258572726051352453113, −6.53237373599566920308251467611, −5.89124566793935794858337844784, −5.58778594917576167237089416214, −5.31847341040317025652784444149, −5.18456852208797076392576149804, −4.41363689023223714955670262308, −4.16124959646750260153692002425, −3.77736647181023331684393977558, −3.40497355091257219311234723298, −2.68497131524621931696567812514, −2.51893455382646759350120725444, −1.79377042301192777583866116513, −1.31442151736057535553137412283, 0, 0,
1.31442151736057535553137412283, 1.79377042301192777583866116513, 2.51893455382646759350120725444, 2.68497131524621931696567812514, 3.40497355091257219311234723298, 3.77736647181023331684393977558, 4.16124959646750260153692002425, 4.41363689023223714955670262308, 5.18456852208797076392576149804, 5.31847341040317025652784444149, 5.58778594917576167237089416214, 5.89124566793935794858337844784, 6.53237373599566920308251467611, 6.69946625258572726051352453113, 6.85168186746843464535776811980, 7.27620223359216700117805352198, 7.939959566335257094233292334813, 8.066233460339454264682989348086
