| L(s) = 1 | + 2·3-s − 3·5-s + 3·9-s + 11-s − 5·13-s − 6·15-s + 2·17-s − 3·19-s − 9·23-s + 25-s + 4·27-s − 2·31-s + 2·33-s + 2·37-s − 10·39-s − 3·41-s + 3·43-s − 9·45-s − 14·47-s − 14·49-s + 4·51-s − 8·53-s − 3·55-s − 6·57-s − 6·59-s − 10·61-s + 15·65-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 1.34·5-s + 9-s + 0.301·11-s − 1.38·13-s − 1.54·15-s + 0.485·17-s − 0.688·19-s − 1.87·23-s + 1/5·25-s + 0.769·27-s − 0.359·31-s + 0.348·33-s + 0.328·37-s − 1.60·39-s − 0.468·41-s + 0.457·43-s − 1.34·45-s − 2.04·47-s − 2·49-s + 0.560·51-s − 1.09·53-s − 0.404·55-s − 0.794·57-s − 0.781·59-s − 1.28·61-s + 1.86·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10653696 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10653696 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 17 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + 3 T + 8 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $D_{4}$ | \( 1 - T + 18 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 5 T + 28 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 3 T + 2 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 9 T + 62 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 2 T + 46 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 2 T + 58 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 3 T + 80 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 3 T + 50 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 14 T + 126 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 8 T + 54 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 6 T + 110 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 10 T + 130 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $D_{4}$ | \( 1 - 4 T + 78 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 8 T + 94 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 6 T + 14 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 10 T + 174 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 6 T + 170 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 14 T + 226 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
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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
−8.175387349307092330866432525733, −7.976239346210505347900831599626, −7.88245017407199102278922929605, −7.66958124429445310156114793500, −7.00471766310461791530075055116, −6.80731318701464892773369874759, −6.14050597117129244922367414646, −6.08315293793857381057948483143, −5.21474058298934090030986704003, −4.83090479900545377165406054620, −4.45880239605576722167619747664, −4.15514387858947700478349899579, −3.68092421750048638947815465144, −3.25975499799086562428478505440, −3.00252034930166978726310266312, −2.30250733017297236854970709578, −1.84943934846389290216637903309, −1.40364878495723912244449059606, 0, 0,
1.40364878495723912244449059606, 1.84943934846389290216637903309, 2.30250733017297236854970709578, 3.00252034930166978726310266312, 3.25975499799086562428478505440, 3.68092421750048638947815465144, 4.15514387858947700478349899579, 4.45880239605576722167619747664, 4.83090479900545377165406054620, 5.21474058298934090030986704003, 6.08315293793857381057948483143, 6.14050597117129244922367414646, 6.80731318701464892773369874759, 7.00471766310461791530075055116, 7.66958124429445310156114793500, 7.88245017407199102278922929605, 7.976239346210505347900831599626, 8.175387349307092330866432525733
