| L(s) = 1 | − 2·3-s + 2·5-s − 7-s + 3·9-s − 7·11-s − 4·13-s − 4·15-s + 2·17-s + 7·19-s + 2·21-s − 23-s + 3·25-s − 4·27-s − 6·29-s + 2·31-s + 14·33-s − 2·35-s + 10·37-s + 8·39-s − 10·41-s − 7·43-s + 6·45-s + 2·47-s − 5·49-s − 4·51-s − 3·53-s − 14·55-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 0.894·5-s − 0.377·7-s + 9-s − 2.11·11-s − 1.10·13-s − 1.03·15-s + 0.485·17-s + 1.60·19-s + 0.436·21-s − 0.208·23-s + 3/5·25-s − 0.769·27-s − 1.11·29-s + 0.359·31-s + 2.43·33-s − 0.338·35-s + 1.64·37-s + 1.28·39-s − 1.56·41-s − 1.06·43-s + 0.894·45-s + 0.291·47-s − 5/7·49-s − 0.560·51-s − 0.412·53-s − 1.88·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 55353600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 55353600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.400978446\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.400978446\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 31 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 7 | $D_{4}$ | \( 1 + T + 6 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 7 T + 26 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 7 T + 42 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + T + 38 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 6 T + 34 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 10 T + 66 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 10 T + 74 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 7 T + 90 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 2 T + 62 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 3 T + 100 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 6 T + 94 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 10 T + 126 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + T + 134 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - T + 72 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 15 T + 206 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 89 | $D_{4}$ | \( 1 - 3 T + 172 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 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
−7.932808906295632104561008388888, −7.66783905949348336052332401885, −7.35759649617133085084213610067, −7.10456628294832911001876840562, −6.44156729180421202560633581325, −6.40524094479749913293433631863, −5.91031111629671526535015958420, −5.52291455304422027370428767041, −5.21632208427655927573190629761, −5.08473638058617748741662566785, −4.83407893146292857159888524072, −4.34028292660842255750717023413, −3.59110078042870576215247492418, −3.36699608642799352937235740028, −2.73251057858034303133200299409, −2.63040779039133934182153000648, −1.86643750655640715952638526395, −1.68642616142175905928125675151, −0.73385146388857122456868724997, −0.42322978494496860810746944405,
0.42322978494496860810746944405, 0.73385146388857122456868724997, 1.68642616142175905928125675151, 1.86643750655640715952638526395, 2.63040779039133934182153000648, 2.73251057858034303133200299409, 3.36699608642799352937235740028, 3.59110078042870576215247492418, 4.34028292660842255750717023413, 4.83407893146292857159888524072, 5.08473638058617748741662566785, 5.21632208427655927573190629761, 5.52291455304422027370428767041, 5.91031111629671526535015958420, 6.40524094479749913293433631863, 6.44156729180421202560633581325, 7.10456628294832911001876840562, 7.35759649617133085084213610067, 7.66783905949348336052332401885, 7.932808906295632104561008388888
