| L(s) = 1 | + 2·5-s − 2·7-s + 2·9-s − 8·11-s + 2·13-s − 4·17-s + 8·19-s + 8·23-s + 3·25-s + 12·29-s + 8·31-s − 4·35-s + 4·37-s − 4·41-s + 4·45-s + 3·49-s + 12·53-s − 16·55-s − 8·59-s − 12·61-s − 4·63-s + 4·65-s + 8·67-s − 16·71-s + 12·73-s + 16·77-s − 5·81-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 0.755·7-s + 2/3·9-s − 2.41·11-s + 0.554·13-s − 0.970·17-s + 1.83·19-s + 1.66·23-s + 3/5·25-s + 2.22·29-s + 1.43·31-s − 0.676·35-s + 0.657·37-s − 0.624·41-s + 0.596·45-s + 3/7·49-s + 1.64·53-s − 2.15·55-s − 1.04·59-s − 1.53·61-s − 0.503·63-s + 0.496·65-s + 0.977·67-s − 1.89·71-s + 1.40·73-s + 1.82·77-s − 5/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 52998400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 52998400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.890605727\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.890605727\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 7 | $C_1$ | \( ( 1 + T )^{2} \) |
| 13 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 17 | $D_{4}$ | \( 1 + 4 T + 30 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 8 T + 46 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 8 T + 54 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $D_{4}$ | \( 1 - 8 T + 46 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 4 T + 46 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 4 T + 14 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 42 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 12 T + 134 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 8 T + 126 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 12 T + 126 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 8 T + 118 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 16 T + 198 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 12 T + 150 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 16 T + 198 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 28 T + 366 T^{2} - 28 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 12 T + 102 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
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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
−8.014024767934606547954354253573, −7.65528396808608804236421481086, −7.46359662234853616978496560907, −7.02673862678773194942875220204, −6.57655514032561892287641293779, −6.46248139657619844299389902742, −5.98592149011740776196442977598, −5.69829169567900237107803570424, −5.10689268017440755531100838397, −4.88480190790594557117897763258, −4.82840469214355588118969599284, −4.31079810338088371190190276284, −3.51662683075383156814036146202, −3.26669710906927377418460741795, −2.77644895753081758617277618141, −2.62678786718903676275713152351, −2.24797792380327376720337753192, −1.45905222776956526013392719557, −0.922505052848828195490066298557, −0.58510382046467137399007286378,
0.58510382046467137399007286378, 0.922505052848828195490066298557, 1.45905222776956526013392719557, 2.24797792380327376720337753192, 2.62678786718903676275713152351, 2.77644895753081758617277618141, 3.26669710906927377418460741795, 3.51662683075383156814036146202, 4.31079810338088371190190276284, 4.82840469214355588118969599284, 4.88480190790594557117897763258, 5.10689268017440755531100838397, 5.69829169567900237107803570424, 5.98592149011740776196442977598, 6.46248139657619844299389902742, 6.57655514032561892287641293779, 7.02673862678773194942875220204, 7.46359662234853616978496560907, 7.65528396808608804236421481086, 8.014024767934606547954354253573
