| L(s) = 1 | − 2·5-s + 2·7-s − 4·13-s + 6·17-s − 2·19-s − 4·25-s + 2·29-s + 4·31-s − 4·35-s − 20·37-s + 4·41-s − 4·43-s + 6·47-s + 3·49-s + 10·53-s − 16·59-s − 4·61-s + 8·65-s − 20·67-s − 6·71-s + 8·79-s − 10·83-s − 12·85-s + 4·89-s − 8·91-s + 4·95-s + 4·97-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 0.755·7-s − 1.10·13-s + 1.45·17-s − 0.458·19-s − 4/5·25-s + 0.371·29-s + 0.718·31-s − 0.676·35-s − 3.28·37-s + 0.624·41-s − 0.609·43-s + 0.875·47-s + 3/7·49-s + 1.37·53-s − 2.08·59-s − 0.512·61-s + 0.992·65-s − 2.44·67-s − 0.712·71-s + 0.900·79-s − 1.09·83-s − 1.30·85-s + 0.423·89-s − 0.838·91-s + 0.410·95-s + 0.406·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 91699776 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91699776 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
| 19 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 5 | $D_{4}$ | \( 1 + 2 T + 8 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $D_{4}$ | \( 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 6 T + 40 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 41 | $D_{4}$ | \( 1 - 4 T + 38 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 4 T + 42 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 6 T + 28 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 10 T + 128 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 67 | $D_{4}$ | \( 1 + 20 T + 222 T^{2} + 20 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 6 T + 148 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 83 | $D_{4}$ | \( 1 + 10 T + 164 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 4 T + 134 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
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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
−7.49170059284961769193946417301, −7.41785725743517506545282893079, −6.89530282497592811425562240407, −6.65533759422408064739965805269, −5.97407846117472291393654320493, −5.89171176959547751169034298926, −5.37242293718513507797960455937, −5.11821878806933873201353267539, −4.65559354935670734794959286934, −4.49921340986635616198417796772, −3.98571371761120413035065433667, −3.68216947844877694334366524326, −3.18578486152986762121440700673, −2.97780854710350460558029876933, −2.34721502452652989830253234422, −1.95838419764699571518077665972, −1.43474684391959216273470191158, −1.04872302790162558277440835622, 0, 0,
1.04872302790162558277440835622, 1.43474684391959216273470191158, 1.95838419764699571518077665972, 2.34721502452652989830253234422, 2.97780854710350460558029876933, 3.18578486152986762121440700673, 3.68216947844877694334366524326, 3.98571371761120413035065433667, 4.49921340986635616198417796772, 4.65559354935670734794959286934, 5.11821878806933873201353267539, 5.37242293718513507797960455937, 5.89171176959547751169034298926, 5.97407846117472291393654320493, 6.65533759422408064739965805269, 6.89530282497592811425562240407, 7.41785725743517506545282893079, 7.49170059284961769193946417301
