| L(s) = 1 | + 2·2-s − 2·3-s + 2·4-s − 2·5-s − 4·6-s + 4·7-s + 4·8-s + 3·9-s − 4·10-s + 6·11-s − 4·12-s + 2·13-s + 8·14-s + 4·15-s + 8·16-s + 10·17-s + 6·18-s + 4·19-s − 4·20-s − 8·21-s + 12·22-s − 8·24-s − 4·25-s + 4·26-s − 4·27-s + 8·28-s − 2·29-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 1.15·3-s + 4-s − 0.894·5-s − 1.63·6-s + 1.51·7-s + 1.41·8-s + 9-s − 1.26·10-s + 1.80·11-s − 1.15·12-s + 0.554·13-s + 2.13·14-s + 1.03·15-s + 2·16-s + 2.42·17-s + 1.41·18-s + 0.917·19-s − 0.894·20-s − 1.74·21-s + 2.55·22-s − 1.63·24-s − 4/5·25-s + 0.784·26-s − 0.769·27-s + 1.51·28-s − 0.371·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2518569 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2518569 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.908054317\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.908054317\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_1$ | \( ( 1 + T )^{2} \) |
| 23 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 - p T + p T^{2} - p^{2} T^{3} + p^{2} T^{4} \) |
| 5 | $D_{4}$ | \( 1 + 2 T + 8 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 4 T + 15 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 6 T + 28 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 17 | $D_{4}$ | \( 1 - 10 T + 56 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 4 T + 30 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 2 T - 16 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 37 | $D_{4}$ | \( 1 - 4 T + 51 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 12 T + 106 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 16 T + 147 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 6 T + 76 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 4 T + 62 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 2 T + 44 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 4 T + 99 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 24 T + 275 T^{2} + 24 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 2 T + 116 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 98 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 89 | $D_{4}$ | \( 1 + 6 T + 160 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 16 T + 210 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
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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
−9.710726678137462680201816035408, −9.273722835226276255482589970341, −8.888471249866983017775379926225, −8.034279992757575114953332093505, −7.65462685401747016378657310678, −7.61985594474241822079543889054, −7.46668596655168341563425077685, −6.62611056376319661147003609877, −6.05489698175622977188308942900, −5.88921889767383060529179561493, −5.43750904875789304335230630180, −5.10209092005732472593233544989, −4.40472903806115616935550056854, −4.39917985335952551983996414399, −3.73095764450750418493617157698, −3.69155295897364332914274447322, −2.89923121676509354618852588056, −1.73504804923049686111992196544, −1.30060884479117841379766125597, −1.03062855264119002155536407222,
1.03062855264119002155536407222, 1.30060884479117841379766125597, 1.73504804923049686111992196544, 2.89923121676509354618852588056, 3.69155295897364332914274447322, 3.73095764450750418493617157698, 4.39917985335952551983996414399, 4.40472903806115616935550056854, 5.10209092005732472593233544989, 5.43750904875789304335230630180, 5.88921889767383060529179561493, 6.05489698175622977188308942900, 6.62611056376319661147003609877, 7.46668596655168341563425077685, 7.61985594474241822079543889054, 7.65462685401747016378657310678, 8.034279992757575114953332093505, 8.888471249866983017775379926225, 9.273722835226276255482589970341, 9.710726678137462680201816035408
