| L(s) = 1 | + 4-s − 4·5-s + 2·11-s − 2·13-s − 3·16-s − 2·17-s − 4·19-s − 4·20-s + 4·23-s + 2·25-s − 8·29-s + 10·31-s − 8·37-s − 18·41-s + 16·43-s + 2·44-s + 10·47-s − 2·52-s − 8·53-s − 8·55-s + 2·59-s + 10·61-s − 7·64-s + 8·65-s + 20·67-s − 2·68-s + 12·71-s + ⋯ |
| L(s) = 1 | + 1/2·4-s − 1.78·5-s + 0.603·11-s − 0.554·13-s − 3/4·16-s − 0.485·17-s − 0.917·19-s − 0.894·20-s + 0.834·23-s + 2/5·25-s − 1.48·29-s + 1.79·31-s − 1.31·37-s − 2.81·41-s + 2.43·43-s + 0.301·44-s + 1.45·47-s − 0.277·52-s − 1.09·53-s − 1.07·55-s + 0.260·59-s + 1.28·61-s − 7/8·64-s + 0.992·65-s + 2.44·67-s − 0.242·68-s + 1.42·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23532201 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23532201 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8156848948\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8156848948\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | | \( 1 \) |
| 11 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 13 | $D_{4}$ | \( 1 + 2 T + 22 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 2 T + 30 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 4 T + 22 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 4 T + 30 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_4$ | \( 1 + 8 T + 54 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 10 T + 82 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 8 T + 70 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 18 T + 158 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 47 | $D_{4}$ | \( 1 - 10 T + 114 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 8 T + 102 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 2 T + 114 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 10 T + 142 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 20 T + 214 T^{2} - 20 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 12 T + 158 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 6 T + 150 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 78 T^{2} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 4 T - 10 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 97 | $D_{4}$ | \( 1 + 8 T + 30 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
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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
−8.390489900448908492077675271510, −8.053340879119642523577210272598, −7.74635118263087205985051906570, −7.33736265942462627659716337892, −7.00917811335554784659733751534, −6.65539854500148699915981116015, −6.62362589032368247424947313357, −5.98131456600418421238177895046, −5.45254262090484139526955638860, −5.12057805700518615452876610456, −4.58235911107829137752791272142, −4.38407573056106415985996310991, −3.77464846840460116956361360403, −3.73138349754956685409722701668, −3.28561654963911706295113420025, −2.42545747434980519135615613371, −2.34885027699971876241555197848, −1.76866490657923909616676046718, −0.923587000199556357563959413816, −0.29032680343432583193398379652,
0.29032680343432583193398379652, 0.923587000199556357563959413816, 1.76866490657923909616676046718, 2.34885027699971876241555197848, 2.42545747434980519135615613371, 3.28561654963911706295113420025, 3.73138349754956685409722701668, 3.77464846840460116956361360403, 4.38407573056106415985996310991, 4.58235911107829137752791272142, 5.12057805700518615452876610456, 5.45254262090484139526955638860, 5.98131456600418421238177895046, 6.62362589032368247424947313357, 6.65539854500148699915981116015, 7.00917811335554784659733751534, 7.33736265942462627659716337892, 7.74635118263087205985051906570, 8.053340879119642523577210272598, 8.390489900448908492077675271510
