| L(s) = 1 | − 3-s − 2·7-s + 3·9-s − 7·11-s − 3·13-s − 5·17-s − 2·19-s + 2·21-s + 2·23-s − 8·27-s − 3·29-s + 16·31-s + 7·33-s + 4·37-s + 3·39-s + 2·41-s − 6·43-s + 3·47-s + 3·49-s + 5·51-s − 10·53-s + 2·57-s − 16·59-s + 6·61-s − 6·63-s − 8·67-s − 2·69-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.755·7-s + 9-s − 2.11·11-s − 0.832·13-s − 1.21·17-s − 0.458·19-s + 0.436·21-s + 0.417·23-s − 1.53·27-s − 0.557·29-s + 2.87·31-s + 1.21·33-s + 0.657·37-s + 0.480·39-s + 0.312·41-s − 0.914·43-s + 0.437·47-s + 3/7·49-s + 0.700·51-s − 1.37·53-s + 0.264·57-s − 2.08·59-s + 0.768·61-s − 0.755·63-s − 0.977·67-s − 0.240·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7840000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7840000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + T - 2 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 7 T + 26 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 3 T + 20 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 5 T + 32 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 2 T + 6 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 2 T + 14 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 3 T + 52 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $D_{4}$ | \( 1 - 2 T + 50 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 6 T + 62 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 3 T + 22 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 10 T + 98 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 61 | $D_{4}$ | \( 1 - 6 T + 98 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 79 | $D_{4}$ | \( 1 + 13 T + 126 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 4 T + 38 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 18 T + 226 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 9 T + 8 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
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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
−8.469385746175580600078341961372, −8.224722954226784453582741987439, −7.76821946477161216257359517067, −7.34845541634258764100941244028, −7.32340715727196821072620736367, −6.50414034821844836720647404901, −6.36421937294974870428384319260, −6.11524050565479337940016802101, −5.35012547877116992995582514307, −5.20361005409404579303554418881, −4.61166037926019937927320229382, −4.44206733412735850198227956170, −4.02007344237616772428912601475, −3.14780636996654605442263346659, −2.83895993711404022172942157477, −2.46051753898991962673161795865, −1.91254589019427177590593686500, −1.11708032024989274593662038403, 0, 0,
1.11708032024989274593662038403, 1.91254589019427177590593686500, 2.46051753898991962673161795865, 2.83895993711404022172942157477, 3.14780636996654605442263346659, 4.02007344237616772428912601475, 4.44206733412735850198227956170, 4.61166037926019937927320229382, 5.20361005409404579303554418881, 5.35012547877116992995582514307, 6.11524050565479337940016802101, 6.36421937294974870428384319260, 6.50414034821844836720647404901, 7.32340715727196821072620736367, 7.34845541634258764100941244028, 7.76821946477161216257359517067, 8.224722954226784453582741987439, 8.469385746175580600078341961372
