| L(s) = 1 | + 2·3-s + 3·9-s − 2·13-s − 2·17-s − 4·23-s + 4·27-s − 6·29-s − 8·31-s + 8·37-s − 4·39-s − 12·41-s − 4·43-s − 4·47-s − 11·49-s − 4·51-s + 10·53-s − 18·61-s − 12·67-s − 8·69-s − 8·71-s + 28·79-s + 5·81-s − 24·83-s − 12·87-s + 8·89-s − 16·93-s + 4·97-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 9-s − 0.554·13-s − 0.485·17-s − 0.834·23-s + 0.769·27-s − 1.11·29-s − 1.43·31-s + 1.31·37-s − 0.640·39-s − 1.87·41-s − 0.609·43-s − 0.583·47-s − 1.57·49-s − 0.560·51-s + 1.37·53-s − 2.30·61-s − 1.46·67-s − 0.963·69-s − 0.949·71-s + 3.15·79-s + 5/9·81-s − 2.63·83-s − 1.28·87-s + 0.847·89-s − 1.65·93-s + 0.406·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 60840000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60840000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 5 | | \( 1 \) |
| 13 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 7 | $C_2^2$ | \( 1 + 11 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 19 T^{2} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 2 T + 23 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 29 | $D_{4}$ | \( 1 + 6 T + 19 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 8 T + 51 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 8 T + 78 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 12 T + 106 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 4 T + 42 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 4 T + 23 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 10 T + 83 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 115 T^{2} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 18 T + 191 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 12 T + 167 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 8 T + 50 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 134 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 83 | $D_{4}$ | \( 1 + 24 T + 307 T^{2} + 24 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 8 T + 146 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 4 T + 150 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
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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
−7.81931526795431775030148457717, −7.48961279535085172827033369351, −6.93076419460991232522066732014, −6.87778309291198541974304450250, −6.21290191906727633118859970872, −6.16917976653034522961243962019, −5.47009164199701246073051548577, −5.25818093304295091329302169154, −4.70976069931090867610643054847, −4.53942545429798153325837382977, −3.90137553403901883946351988471, −3.79553154389294020237737901574, −3.18157302446985698821435014933, −3.06265860163795337941528915294, −2.29206969083974955684692140418, −2.21420271529053899329789591262, −1.49714577594618275039195659693, −1.39149897169086584558767442574, 0, 0,
1.39149897169086584558767442574, 1.49714577594618275039195659693, 2.21420271529053899329789591262, 2.29206969083974955684692140418, 3.06265860163795337941528915294, 3.18157302446985698821435014933, 3.79553154389294020237737901574, 3.90137553403901883946351988471, 4.53942545429798153325837382977, 4.70976069931090867610643054847, 5.25818093304295091329302169154, 5.47009164199701246073051548577, 6.16917976653034522961243962019, 6.21290191906727633118859970872, 6.87778309291198541974304450250, 6.93076419460991232522066732014, 7.48961279535085172827033369351, 7.81931526795431775030148457717
