| L(s) = 1 | − 4·7-s + 12·17-s − 8·23-s − 25-s − 20·31-s − 20·41-s − 8·47-s − 2·49-s − 8·71-s + 20·73-s + 28·79-s − 28·89-s − 20·97-s + 4·103-s − 12·113-s − 48·119-s + 6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 32·161-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | − 1.51·7-s + 2.91·17-s − 1.66·23-s − 1/5·25-s − 3.59·31-s − 3.12·41-s − 1.16·47-s − 2/7·49-s − 0.949·71-s + 2.34·73-s + 3.15·79-s − 2.96·89-s − 2.03·97-s + 0.394·103-s − 1.12·113-s − 4.40·119-s + 6/11·121-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 + 2.52·161-s + 0.0783·163-s + 0.0773·167-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2073600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2073600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7189588076\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7189588076\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| good | 7 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 54 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
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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
−9.821829431708892956964254494006, −9.354094253682609085681039595783, −9.255104980425820087005714032841, −8.289424434526591841867225924358, −8.138988632522686263166671829296, −7.86282324019680072946234411252, −7.24466066830627603973922874964, −6.76630783323006265674513399981, −6.67565302581894547538290429185, −5.81800693973327795464505440020, −5.72337206167562305029026070245, −5.30241617411372266717401701421, −4.85440100223836599974660812731, −3.77902288694413270098235946716, −3.69490182960377994125609730582, −3.36046617832412111145231734701, −2.90385070972141966395465254389, −1.81808407337671702396876871288, −1.63036692113692049996276112208, −0.33172000155736559573941842711,
0.33172000155736559573941842711, 1.63036692113692049996276112208, 1.81808407337671702396876871288, 2.90385070972141966395465254389, 3.36046617832412111145231734701, 3.69490182960377994125609730582, 3.77902288694413270098235946716, 4.85440100223836599974660812731, 5.30241617411372266717401701421, 5.72337206167562305029026070245, 5.81800693973327795464505440020, 6.67565302581894547538290429185, 6.76630783323006265674513399981, 7.24466066830627603973922874964, 7.86282324019680072946234411252, 8.138988632522686263166671829296, 8.289424434526591841867225924358, 9.255104980425820087005714032841, 9.354094253682609085681039595783, 9.821829431708892956964254494006
