| L(s) = 1 | + 2·3-s − 2·7-s + 3·9-s + 2·11-s + 2·13-s − 6·17-s − 4·21-s + 4·27-s + 16·29-s + 4·31-s + 4·33-s − 8·37-s + 4·39-s + 16·41-s + 14·43-s − 4·47-s + 2·49-s − 12·51-s − 4·53-s + 16·59-s − 6·63-s + 8·67-s + 6·73-s − 4·77-s − 8·79-s + 5·81-s + 14·83-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.755·7-s + 9-s + 0.603·11-s + 0.554·13-s − 1.45·17-s − 0.872·21-s + 0.769·27-s + 2.97·29-s + 0.718·31-s + 0.696·33-s − 1.31·37-s + 0.640·39-s + 2.49·41-s + 2.13·43-s − 0.583·47-s + 2/7·49-s − 1.68·51-s − 0.549·53-s + 2.08·59-s − 0.755·63-s + 0.977·67-s + 0.702·73-s − 0.455·77-s − 0.900·79-s + 5/9·81-s + 1.53·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10890000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10890000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.970644369\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.970644369\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 5 | | \( 1 \) |
| 11 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 7 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 2 T + 14 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 6 T + 30 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 31 | $D_{4}$ | \( 1 - 4 T + 14 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 8 T + 38 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 43 | $D_{4}$ | \( 1 - 14 T + 122 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 4 T + 46 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 + 70 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 66 T^{2} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 6 T + 142 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 8 T + 122 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 14 T + 202 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $D_{4}$ | \( 1 + 4 T - 10 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
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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
−8.813916005310831480734545076274, −8.589466895274595495270014622432, −8.104871630864061629619481808515, −7.84272339484630527654978281856, −7.37224559782366778563036437549, −6.85971671724667968473629109763, −6.53590375197783792154743449792, −6.45434669837707989332021700025, −6.00477281524175025567788726305, −5.34834910600729372796602972438, −4.89895432170731489937035486542, −4.37912285316047096388157517334, −4.02758490704729885785208751627, −3.89646430417167543525815117589, −3.02057345058093783607344521113, −2.95083270163911374555007857731, −2.32153715271758619603875554053, −2.04804689660414530612536560768, −1.04151831079557353750362355826, −0.75645246994083338533263384375,
0.75645246994083338533263384375, 1.04151831079557353750362355826, 2.04804689660414530612536560768, 2.32153715271758619603875554053, 2.95083270163911374555007857731, 3.02057345058093783607344521113, 3.89646430417167543525815117589, 4.02758490704729885785208751627, 4.37912285316047096388157517334, 4.89895432170731489937035486542, 5.34834910600729372796602972438, 6.00477281524175025567788726305, 6.45434669837707989332021700025, 6.53590375197783792154743449792, 6.85971671724667968473629109763, 7.37224559782366778563036437549, 7.84272339484630527654978281856, 8.104871630864061629619481808515, 8.589466895274595495270014622432, 8.813916005310831480734545076274
