| L(s) = 1 | + 2·3-s + 2·5-s − 7-s + 3·9-s − 11-s + 4·13-s + 4·15-s + 2·17-s + 3·19-s − 2·21-s − 7·23-s + 3·25-s + 4·27-s + 6·29-s − 2·31-s − 2·33-s − 2·35-s + 2·37-s + 8·39-s + 6·41-s − 5·43-s + 6·45-s − 2·47-s − 9·49-s + 4·51-s + 19·53-s − 2·55-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 0.894·5-s − 0.377·7-s + 9-s − 0.301·11-s + 1.10·13-s + 1.03·15-s + 0.485·17-s + 0.688·19-s − 0.436·21-s − 1.45·23-s + 3/5·25-s + 0.769·27-s + 1.11·29-s − 0.359·31-s − 0.348·33-s − 0.338·35-s + 0.328·37-s + 1.28·39-s + 0.937·41-s − 0.762·43-s + 0.894·45-s − 0.291·47-s − 9/7·49-s + 0.560·51-s + 2.60·53-s − 0.269·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 55353600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 55353600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(8.108565189\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.108565189\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 31 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 7 | $D_{4}$ | \( 1 + T + 10 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + T + 18 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $D_{4}$ | \( 1 - 2 T + 18 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 3 T + 2 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 7 T + 54 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 6 T + 50 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 2 T + 58 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 6 T + 74 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 5 T - 14 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 2 T + 78 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 19 T + 192 T^{2} - 19 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 18 T + 182 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 67 | $D_{4}$ | \( 1 - 6 T - 10 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 3 T + 106 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + T + 40 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 21 T + 230 T^{2} - 21 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 12 T + 134 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 15 T + 196 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
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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
−8.102653435309411881512184918512, −7.953679186914090726885351048719, −7.36637808857125504513710442176, −6.91842935713703258578292829214, −6.81661461447846374973625135662, −6.40942840968480926456271871619, −5.88835378692802499152407847734, −5.72538473438087726042693027649, −5.20995794693537539466641240697, −5.09841851498645150460449934458, −4.22106588411592074670113032366, −4.12186358114590342152312771630, −3.68984532520236152109015142385, −3.31749572217411944440369375061, −2.84662738754063065505000345860, −2.56178389189696991677469835605, −1.92370384464123265147536759049, −1.86377409912771564889771108507, −0.953577815886392368258272183956, −0.74490729290079258126477942500,
0.74490729290079258126477942500, 0.953577815886392368258272183956, 1.86377409912771564889771108507, 1.92370384464123265147536759049, 2.56178389189696991677469835605, 2.84662738754063065505000345860, 3.31749572217411944440369375061, 3.68984532520236152109015142385, 4.12186358114590342152312771630, 4.22106588411592074670113032366, 5.09841851498645150460449934458, 5.20995794693537539466641240697, 5.72538473438087726042693027649, 5.88835378692802499152407847734, 6.40942840968480926456271871619, 6.81661461447846374973625135662, 6.91842935713703258578292829214, 7.36637808857125504513710442176, 7.953679186914090726885351048719, 8.102653435309411881512184918512
