| L(s) = 1 | − 2·5-s + 2·13-s + 14·19-s + 3·25-s + 12·29-s + 2·31-s − 2·37-s − 2·43-s − 12·47-s − 12·59-s + 8·61-s − 4·65-s − 2·67-s − 10·73-s + 22·79-s − 12·83-s − 12·89-s − 28·95-s − 16·97-s − 24·101-s + 26·103-s − 2·109-s + 36·113-s − 4·121-s − 4·125-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 0.554·13-s + 3.21·19-s + 3/5·25-s + 2.22·29-s + 0.359·31-s − 0.328·37-s − 0.304·43-s − 1.75·47-s − 1.56·59-s + 1.02·61-s − 0.496·65-s − 0.244·67-s − 1.17·73-s + 2.47·79-s − 1.31·83-s − 1.27·89-s − 2.87·95-s − 1.62·97-s − 2.38·101-s + 2.56·103-s − 0.191·109-s + 3.38·113-s − 0.363·121-s − 0.357·125-s + 0.0887·127-s + 0.0873·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 77792400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 77792400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.543420924\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.543420924\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 7 | | \( 1 \) |
| good | 11 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 2 T + 9 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 16 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 28 T^{2} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 12 T + 76 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 2 T - 9 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 2 T + 57 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 64 T^{2} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 2 T + 69 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 12 T + 136 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 8 T + 66 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 2 T + 117 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 20 T^{2} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 10 T + 153 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2} \) |
| 83 | $D_{4}$ | \( 1 + 12 T + 184 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 12 T + 196 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 16 T + 186 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
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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
−7.80299411096270061986814118464, −7.75854169192619027655371942108, −7.11028074128686601255734189813, −7.08393657925861778611405906491, −6.50590356253095170190885934988, −6.43368215861921039141253124270, −5.67623828617640359292127385366, −5.60058961235877587276937059915, −5.06577986215996851387848864635, −4.86341816004139457962369503023, −4.35480373154367928825342415078, −4.18070680872129161695802889610, −3.45512994672663051644891800793, −3.25927010456485270974687211619, −2.96072215036706702734488614926, −2.75191700138546762487572789827, −1.72247674039738335076559472967, −1.53327493052458161839075221660, −0.798282995829753239027675648664, −0.60611729130721679839025420884,
0.60611729130721679839025420884, 0.798282995829753239027675648664, 1.53327493052458161839075221660, 1.72247674039738335076559472967, 2.75191700138546762487572789827, 2.96072215036706702734488614926, 3.25927010456485270974687211619, 3.45512994672663051644891800793, 4.18070680872129161695802889610, 4.35480373154367928825342415078, 4.86341816004139457962369503023, 5.06577986215996851387848864635, 5.60058961235877587276937059915, 5.67623828617640359292127385366, 6.43368215861921039141253124270, 6.50590356253095170190885934988, 7.08393657925861778611405906491, 7.11028074128686601255734189813, 7.75854169192619027655371942108, 7.80299411096270061986814118464
