| 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
−8.186984920563681212224079960564, −8.113754606388793963429442659203, −7.54628899569875679202342695532, −7.38058949951087023300280243331, −6.96751593213437349746143827228, −6.79440782291064913632106206197, −6.16554163456093393270035191951, −5.75556725315195633973748696062, −5.29968725353870906610079705920, −5.01916057239303604203829220298, −4.66347472290807897078471778961, −4.35662704594242148774155863413, −3.73114903292548287166216394417, −3.42290107452167996677182955413, −2.70089089520428647701352971308, −2.58371630432263621335289494690, −1.38929607247975927588987815895, −1.15631390177786704923644781862, 0, 0,
1.15631390177786704923644781862, 1.38929607247975927588987815895, 2.58371630432263621335289494690, 2.70089089520428647701352971308, 3.42290107452167996677182955413, 3.73114903292548287166216394417, 4.35662704594242148774155863413, 4.66347472290807897078471778961, 5.01916057239303604203829220298, 5.29968725353870906610079705920, 5.75556725315195633973748696062, 6.16554163456093393270035191951, 6.79440782291064913632106206197, 6.96751593213437349746143827228, 7.38058949951087023300280243331, 7.54628899569875679202342695532, 8.113754606388793963429442659203, 8.186984920563681212224079960564
