| L(s) = 1 | + 2·2-s + 3·4-s + 2·7-s + 4·8-s − 3·11-s − 2·13-s + 4·14-s + 5·16-s − 17-s + 7·19-s − 6·22-s + 3·23-s − 4·26-s + 6·28-s − 3·29-s + 7·31-s + 6·32-s − 2·34-s − 2·37-s + 14·38-s − 11·41-s + 14·43-s − 9·44-s + 6·46-s + 11·47-s + 3·49-s − 6·52-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 3/2·4-s + 0.755·7-s + 1.41·8-s − 0.904·11-s − 0.554·13-s + 1.06·14-s + 5/4·16-s − 0.242·17-s + 1.60·19-s − 1.27·22-s + 0.625·23-s − 0.784·26-s + 1.13·28-s − 0.557·29-s + 1.25·31-s + 1.06·32-s − 0.342·34-s − 0.328·37-s + 2.27·38-s − 1.71·41-s + 2.13·43-s − 1.35·44-s + 0.884·46-s + 1.60·47-s + 3/7·49-s − 0.832·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 89302500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 89302500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(11.85105394\) |
| \(L(\frac12)\) |
\(\approx\) |
\(11.85105394\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 11 | $D_{4}$ | \( 1 + 3 T + 3 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 17 | $D_{4}$ | \( 1 + T + 10 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 7 T + 26 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 3 T + 24 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 3 T + 36 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 7 T + 50 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 2 T - 22 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 11 T + 88 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 47 | $D_{4}$ | \( 1 - 11 T + 100 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 59 | $D_{4}$ | \( 1 + 5 T + 100 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 2 T + 26 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 71 | $D_{4}$ | \( 1 + T + 118 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 5 T + 128 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $D_{4}$ | \( 1 - 25 T + 298 T^{2} - 25 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
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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
−7.54397630659359637958543321072, −7.45894900413363941417494666452, −7.24288785383792346944843140716, −6.92965447657804362721612024298, −6.34058432427516505542461850455, −6.10389681517277424344333309978, −5.55773068564983543297440822946, −5.48921867524837379561934437223, −5.01104720498559807877775797200, −4.91399825874675983251210924876, −4.41558405753447315614809621499, −4.16504689070533770961172529876, −3.55188127574179503479569942350, −3.38190738018846224212386274471, −2.75896154154730733505326172919, −2.62198351390060116127385880035, −2.07790059367656303941126749460, −1.77241886556812488301615457582, −0.930753800341199878016630387161, −0.69693807153555354229597822934,
0.69693807153555354229597822934, 0.930753800341199878016630387161, 1.77241886556812488301615457582, 2.07790059367656303941126749460, 2.62198351390060116127385880035, 2.75896154154730733505326172919, 3.38190738018846224212386274471, 3.55188127574179503479569942350, 4.16504689070533770961172529876, 4.41558405753447315614809621499, 4.91399825874675983251210924876, 5.01104720498559807877775797200, 5.48921867524837379561934437223, 5.55773068564983543297440822946, 6.10389681517277424344333309978, 6.34058432427516505542461850455, 6.92965447657804362721612024298, 7.24288785383792346944843140716, 7.45894900413363941417494666452, 7.54397630659359637958543321072
