| L(s) = 1 | − 2·2-s − 2·3-s + 3·4-s + 4·6-s − 4·8-s + 3·9-s + 4·11-s − 6·12-s + 5·16-s − 6·18-s − 8·22-s − 4·23-s + 8·24-s − 4·27-s + 8·29-s − 4·31-s − 6·32-s − 8·33-s + 9·36-s − 12·41-s − 12·43-s + 12·44-s + 8·46-s + 4·47-s − 10·48-s − 4·53-s + 8·54-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 1.15·3-s + 3/2·4-s + 1.63·6-s − 1.41·8-s + 9-s + 1.20·11-s − 1.73·12-s + 5/4·16-s − 1.41·18-s − 1.70·22-s − 0.834·23-s + 1.63·24-s − 0.769·27-s + 1.48·29-s − 0.718·31-s − 1.06·32-s − 1.39·33-s + 3/2·36-s − 1.87·41-s − 1.82·43-s + 1.80·44-s + 1.17·46-s + 0.583·47-s − 1.44·48-s − 0.549·53-s + 1.08·54-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 54022500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 54022500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 5 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 11 | $D_{4}$ | \( 1 - 4 T + 24 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 32 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 23 | $C_4$ | \( 1 + 4 T + 42 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 8 T + 56 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 4 T + 16 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 72 T^{2} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 12 T + 110 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 12 T + 120 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 4 T + 48 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 4 T - 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 12 T + 82 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 12 T + 126 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 4 T + 40 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 16 T + 198 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 4 T + 118 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 16 T + 222 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 4 T + 110 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 12 T + 222 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
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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
−7.65202159197811897661771772486, −7.56682112840938948521383312087, −6.80111516454368882120631967970, −6.70678610074099557104642435046, −6.48021240206211822825879162667, −6.32762500768754890516844744827, −5.64263348248367000894460420994, −5.47210269890942041642994780735, −4.91574593646382640985574105025, −4.69060376270620398712389995491, −4.06384415691686472282312807218, −3.79871874877343957848259467762, −3.20715369919979125672891039939, −2.96284458867126933233507779539, −2.08259469732522010711152812768, −1.88746836558502288294215241965, −1.22153912607198122643994745514, −1.10554736526908233467887567660, 0, 0,
1.10554736526908233467887567660, 1.22153912607198122643994745514, 1.88746836558502288294215241965, 2.08259469732522010711152812768, 2.96284458867126933233507779539, 3.20715369919979125672891039939, 3.79871874877343957848259467762, 4.06384415691686472282312807218, 4.69060376270620398712389995491, 4.91574593646382640985574105025, 5.47210269890942041642994780735, 5.64263348248367000894460420994, 6.32762500768754890516844744827, 6.48021240206211822825879162667, 6.70678610074099557104642435046, 6.80111516454368882120631967970, 7.56682112840938948521383312087, 7.65202159197811897661771772486
