| L(s) = 1 | + 4·5-s + 2·7-s + 4·11-s − 13-s + 4·17-s − 4·19-s + 5·25-s + 6·29-s + 10·31-s + 8·35-s + 20·37-s − 8·41-s − 4·43-s + 4·47-s + 7·49-s − 20·53-s + 16·55-s + 8·59-s + 14·61-s − 4·65-s − 2·67-s + 32·71-s − 20·73-s + 8·77-s + 16·79-s + 16·85-s − 8·89-s + ⋯ |
| L(s) = 1 | + 1.78·5-s + 0.755·7-s + 1.20·11-s − 0.277·13-s + 0.970·17-s − 0.917·19-s + 25-s + 1.11·29-s + 1.79·31-s + 1.35·35-s + 3.28·37-s − 1.24·41-s − 0.609·43-s + 0.583·47-s + 49-s − 2.74·53-s + 2.15·55-s + 1.04·59-s + 1.79·61-s − 0.496·65-s − 0.244·67-s + 3.79·71-s − 2.34·73-s + 0.911·77-s + 1.80·79-s + 1.73·85-s − 0.847·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 17740944 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17740944 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.865208568\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.865208568\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 13 | $C_2$ | \( 1 + T + T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - 4 T + 11 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 2 T - 3 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 4 T + 5 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 6 T + 7 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 10 T + 69 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 8 T + 23 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 4 T - 27 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 4 T - 31 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 8 T + 5 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + 2 T - 63 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 16 T + 177 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 2 T - 93 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
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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
−8.426307134069867592713667197275, −8.318722787635579657883370759815, −7.894394610814297621259935060266, −7.67760689707538305969090227413, −6.78851364991357310204536516151, −6.75387822901666802676019035979, −6.35999163851711778941089101648, −6.05129178862079530266475343784, −5.72381874997567182015928709245, −5.30923009211381278515036037747, −4.69407514231211404461411533710, −4.68283184720550380875812724769, −4.11854933526680175591099442388, −3.62878154429532031765140938907, −3.02486775436371686068199275395, −2.56371537692863679472990010296, −2.12619821714637538668494324312, −1.81413736804662595977172651594, −0.986201784551478799317341789883, −0.951642609230262267305226911386,
0.951642609230262267305226911386, 0.986201784551478799317341789883, 1.81413736804662595977172651594, 2.12619821714637538668494324312, 2.56371537692863679472990010296, 3.02486775436371686068199275395, 3.62878154429532031765140938907, 4.11854933526680175591099442388, 4.68283184720550380875812724769, 4.69407514231211404461411533710, 5.30923009211381278515036037747, 5.72381874997567182015928709245, 6.05129178862079530266475343784, 6.35999163851711778941089101648, 6.75387822901666802676019035979, 6.78851364991357310204536516151, 7.67760689707538305969090227413, 7.894394610814297621259935060266, 8.318722787635579657883370759815, 8.426307134069867592713667197275
