| L(s) = 1 | − 2-s + 2·3-s − 2·4-s − 2·6-s + 3·8-s − 3·9-s − 4·12-s + 5·13-s + 16-s − 2·17-s + 3·18-s − 9·19-s − 8·23-s + 6·24-s − 5·25-s − 5·26-s − 14·27-s + 9·29-s + 11·31-s − 2·32-s + 2·34-s + 6·36-s − 37-s + 9·38-s + 10·39-s + 7·41-s − 7·43-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.15·3-s − 4-s − 0.816·6-s + 1.06·8-s − 9-s − 1.15·12-s + 1.38·13-s + 1/4·16-s − 0.485·17-s + 0.707·18-s − 2.06·19-s − 1.66·23-s + 1.22·24-s − 25-s − 0.980·26-s − 2.69·27-s + 1.67·29-s + 1.97·31-s − 0.353·32-s + 0.342·34-s + 36-s − 0.164·37-s + 1.45·38-s + 1.60·39-s + 1.09·41-s − 1.06·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 35153041 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35153041 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.261444158\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.261444158\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 7 | | \( 1 \) |
| 11 | | \( 1 \) |
| good | 2 | $D_{4}$ | \( 1 + T + 3 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 3 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 5 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 5 T + 21 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 2 T + 30 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 9 T + 47 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 8 T + 57 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 9 T + 67 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_4$ | \( 1 - 11 T + 91 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + T + 43 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 7 T + 93 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 7 T + 87 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 8 T + 90 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - T + 75 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 6 T + 82 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 4 T + 46 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 19 T + 223 T^{2} - 19 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 6 T + 146 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 15 T + 191 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 12 T + 149 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 89 | $C_4$ | \( 1 - 12 T + 194 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + T + 183 T^{2} + p T^{3} + p^{2} T^{4} \) |
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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
−8.368647448591352857331537795962, −8.193616204334905140652625834012, −7.75150192327009313939181632383, −7.74761967038967721081138168757, −6.63058621092006268686576256166, −6.47670423064896501717624432733, −6.22518632451497074253380647648, −6.00833718744624925692858183887, −5.35019778626958989991761034337, −4.97170742746496682756648585510, −4.38551226835403712344782803131, −4.31579386505978707163035305620, −3.75963851513177812856262787139, −3.50014434679253729856029039893, −3.02743386923758843085507351596, −2.44172110463146504093877165483, −2.10492018620431415828239901451, −1.78269541331589265751300653262, −0.75644194806724836707984506269, −0.42252769540754953989815762702,
0.42252769540754953989815762702, 0.75644194806724836707984506269, 1.78269541331589265751300653262, 2.10492018620431415828239901451, 2.44172110463146504093877165483, 3.02743386923758843085507351596, 3.50014434679253729856029039893, 3.75963851513177812856262787139, 4.31579386505978707163035305620, 4.38551226835403712344782803131, 4.97170742746496682756648585510, 5.35019778626958989991761034337, 6.00833718744624925692858183887, 6.22518632451497074253380647648, 6.47670423064896501717624432733, 6.63058621092006268686576256166, 7.74761967038967721081138168757, 7.75150192327009313939181632383, 8.193616204334905140652625834012, 8.368647448591352857331537795962
