| L(s) = 1 | − 2·5-s − 2·7-s − 4·11-s + 6·13-s − 2·19-s + 8·23-s − 2·25-s − 4·31-s + 4·35-s + 8·41-s − 4·43-s + 12·47-s + 3·49-s + 20·53-s + 8·55-s − 14·59-s + 18·61-s − 12·65-s − 8·67-s + 8·71-s + 12·73-s + 8·77-s − 8·79-s − 14·83-s + 12·89-s − 12·91-s + 4·95-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 0.755·7-s − 1.20·11-s + 1.66·13-s − 0.458·19-s + 1.66·23-s − 2/5·25-s − 0.718·31-s + 0.676·35-s + 1.24·41-s − 0.609·43-s + 1.75·47-s + 3/7·49-s + 2.74·53-s + 1.07·55-s − 1.82·59-s + 2.30·61-s − 1.48·65-s − 0.977·67-s + 0.949·71-s + 1.40·73-s + 0.911·77-s − 0.900·79-s − 1.53·83-s + 1.27·89-s − 1.25·91-s + 0.410·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4064256 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4064256 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.598059836\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.598059836\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 5 | $D_{4}$ | \( 1 + 2 T + 6 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_4$ | \( 1 + 4 T + 6 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 6 T + 30 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 2 T + 34 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 31 | $C_4$ | \( 1 + 4 T + 46 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 54 T^{2} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 8 T + 78 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 4 T + 70 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 12 T + 110 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 59 | $D_{4}$ | \( 1 + 14 T + 162 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 18 T + 198 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $D_{4}$ | \( 1 - 8 T + 78 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 12 T + 102 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 8 T + 94 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 14 T + 210 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $D_{4}$ | \( 1 - 16 T + 238 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
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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
−9.216118930805698326384425623474, −8.840385666258659237197929365376, −8.499819704841678415589022809772, −8.415701441824424481038825910764, −7.57942815225505161296984183868, −7.48870788169394052222953531561, −7.12731850197293463446255077802, −6.65686839378631852179130904464, −5.98462474757423115447068654703, −5.95212335959654055529540080445, −5.36219503203642293123154646832, −4.96462730110672526579612539064, −4.28577434581763831773154501544, −3.99571485946686014180573382297, −3.46281517311672406695435818101, −3.23519203225170933971650768882, −2.49658637187557947599551557296, −2.12079585758260936582977290206, −1.03691613932688361078695266976, −0.54770173636163218077900776414,
0.54770173636163218077900776414, 1.03691613932688361078695266976, 2.12079585758260936582977290206, 2.49658637187557947599551557296, 3.23519203225170933971650768882, 3.46281517311672406695435818101, 3.99571485946686014180573382297, 4.28577434581763831773154501544, 4.96462730110672526579612539064, 5.36219503203642293123154646832, 5.95212335959654055529540080445, 5.98462474757423115447068654703, 6.65686839378631852179130904464, 7.12731850197293463446255077802, 7.48870788169394052222953531561, 7.57942815225505161296984183868, 8.415701441824424481038825910764, 8.499819704841678415589022809772, 8.840385666258659237197929365376, 9.216118930805698326384425623474
