| 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
| \(L(1)\) |
\(\approx\) |
\(3.088803442\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.088803442\) |
| \(L(\frac{3}{2})\) |
|
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
−9.203675513001725535506975781260, −8.868912064327284131099737566901, −8.499163110453923095836011900585, −8.166721805021908766276308092390, −7.86372512401484149104268787148, −7.60215299019775738553685353783, −6.82380813641189119428851670999, −6.72577815600787650516397378463, −6.21199137994958596205717062655, −5.75613912461400587848994784285, −5.43103556783087890696201720351, −4.87151952150274988598379958034, −4.17319197988322784731760768258, −3.98514069108880664016473976885, −3.74041176419834802253792063886, −3.29997066512250909208035383527, −2.16371368149281105888051700463, −2.15158652018762322339335595454, −1.04931903695550526602873363949, −0.807766492554902541597829732893,
0.807766492554902541597829732893, 1.04931903695550526602873363949, 2.15158652018762322339335595454, 2.16371368149281105888051700463, 3.29997066512250909208035383527, 3.74041176419834802253792063886, 3.98514069108880664016473976885, 4.17319197988322784731760768258, 4.87151952150274988598379958034, 5.43103556783087890696201720351, 5.75613912461400587848994784285, 6.21199137994958596205717062655, 6.72577815600787650516397378463, 6.82380813641189119428851670999, 7.60215299019775738553685353783, 7.86372512401484149104268787148, 8.166721805021908766276308092390, 8.499163110453923095836011900585, 8.868912064327284131099737566901, 9.203675513001725535506975781260
