| L(s) = 1 | − 8·7-s − 12·17-s − 16·23-s − 8·25-s − 8·31-s − 4·41-s + 16·47-s + 34·49-s + 20·73-s − 24·79-s + 32·89-s + 16·97-s + 8·103-s − 32·113-s + 96·119-s + 10·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 128·161-s + 163-s + 167-s − 8·169-s + ⋯ |
| L(s) = 1 | − 3.02·7-s − 2.91·17-s − 3.33·23-s − 8/5·25-s − 1.43·31-s − 0.624·41-s + 2.33·47-s + 34/7·49-s + 2.34·73-s − 2.70·79-s + 3.39·89-s + 1.62·97-s + 0.788·103-s − 3.01·113-s + 8.80·119-s + 0.909·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 10.0·161-s + 0.0783·163-s + 0.0773·167-s − 0.615·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 84934656 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84934656 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4110998419\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4110998419\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 + 40 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 72 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 54 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 + 104 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 134 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
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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
−7.66330791894148370073214364521, −7.59114308446760209554129223590, −6.98763763386974586798955879502, −6.93965925733596875526185895906, −6.31149465993820446006633850159, −6.22138687512023386680462486194, −6.07396255644694253825126895823, −5.75275284959785702500702929207, −5.21552013547093499258210996749, −4.68297164842777420597699036065, −4.07442381900264870603532195146, −3.99684416024933049624871436888, −3.68406413907648046145767880545, −3.48164130411424294812662879242, −2.60853005660818765480160873787, −2.54922596404450608833455710750, −1.95971681267844235676834479855, −1.86489216550359917657718624544, −0.44847725867758537800451098283, −0.29008034566186789763933148617,
0.29008034566186789763933148617, 0.44847725867758537800451098283, 1.86489216550359917657718624544, 1.95971681267844235676834479855, 2.54922596404450608833455710750, 2.60853005660818765480160873787, 3.48164130411424294812662879242, 3.68406413907648046145767880545, 3.99684416024933049624871436888, 4.07442381900264870603532195146, 4.68297164842777420597699036065, 5.21552013547093499258210996749, 5.75275284959785702500702929207, 6.07396255644694253825126895823, 6.22138687512023386680462486194, 6.31149465993820446006633850159, 6.93965925733596875526185895906, 6.98763763386974586798955879502, 7.59114308446760209554129223590, 7.66330791894148370073214364521
