| L(s) = 1 | + 2·2-s − 2·3-s + 3·4-s − 4·5-s − 4·6-s + 4·8-s + 3·9-s − 8·10-s + 2·11-s − 6·12-s + 2·13-s + 8·15-s + 5·16-s − 2·17-s + 6·18-s − 10·19-s − 12·20-s + 4·22-s − 8·24-s + 4·25-s + 4·26-s − 4·27-s + 2·29-s + 16·30-s + 6·32-s − 4·33-s − 4·34-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 1.15·3-s + 3/2·4-s − 1.78·5-s − 1.63·6-s + 1.41·8-s + 9-s − 2.52·10-s + 0.603·11-s − 1.73·12-s + 0.554·13-s + 2.06·15-s + 5/4·16-s − 0.485·17-s + 1.41·18-s − 2.29·19-s − 2.68·20-s + 0.852·22-s − 1.63·24-s + 4/5·25-s + 0.784·26-s − 0.769·27-s + 0.371·29-s + 2.92·30-s + 1.06·32-s − 0.696·33-s − 0.685·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14607684 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14607684 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 7 | | \( 1 \) |
| 13 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 5 | $D_{4}$ | \( 1 + 4 T + 12 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 2 T + 21 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 2 T + 33 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 10 T + 55 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 44 T^{2} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 2 T + 51 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 72 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 16 T^{2} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 53 | $D_{4}$ | \( 1 - 2 T + 35 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 10 T + 93 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 6 T + 129 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 2 T + 127 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 144 T^{2} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 12 T + 162 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 4 T + 138 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 8 T + 192 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 16 T + 208 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
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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
−8.113413204190550388226751430637, −7.962251865651287606204258679649, −7.24778851651579104131222721478, −7.19108399046494015444868927495, −6.58905287116292104554837693630, −6.48128066040753662524037487260, −6.07918039980903534585299138554, −5.73954196444726087468153789032, −5.21847147520853028522622065069, −4.76772038847933459310054871518, −4.35810767562569608239089280948, −4.21713916522509687324786626422, −3.73359280838118878535217068790, −3.72826531637788830471425767191, −2.75497545887510073395748543822, −2.55118735194606936406090659544, −1.55303991959244434776358613732, −1.35541431671129224739576996298, 0, 0,
1.35541431671129224739576996298, 1.55303991959244434776358613732, 2.55118735194606936406090659544, 2.75497545887510073395748543822, 3.72826531637788830471425767191, 3.73359280838118878535217068790, 4.21713916522509687324786626422, 4.35810767562569608239089280948, 4.76772038847933459310054871518, 5.21847147520853028522622065069, 5.73954196444726087468153789032, 6.07918039980903534585299138554, 6.48128066040753662524037487260, 6.58905287116292104554837693630, 7.19108399046494015444868927495, 7.24778851651579104131222721478, 7.962251865651287606204258679649, 8.113413204190550388226751430637
