| L(s) = 1 | − 2·5-s + 2·7-s − 4·11-s − 4·13-s + 2·17-s − 2·19-s + 8·23-s + 3·25-s + 4·29-s − 2·31-s − 4·35-s − 18·37-s + 12·41-s − 2·43-s + 14·47-s − 8·49-s + 6·53-s + 8·55-s − 8·59-s − 16·61-s + 8·65-s + 8·67-s − 6·73-s − 8·77-s + 24·79-s − 2·83-s − 4·85-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 0.755·7-s − 1.20·11-s − 1.10·13-s + 0.485·17-s − 0.458·19-s + 1.66·23-s + 3/5·25-s + 0.742·29-s − 0.359·31-s − 0.676·35-s − 2.95·37-s + 1.87·41-s − 0.304·43-s + 2.04·47-s − 8/7·49-s + 0.824·53-s + 1.07·55-s − 1.04·59-s − 2.04·61-s + 0.992·65-s + 0.977·67-s − 0.702·73-s − 0.911·77-s + 2.70·79-s − 0.219·83-s − 0.433·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 41990400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 41990400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.718467964\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.718467964\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 7 | $D_{4}$ | \( 1 - 2 T + 12 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 4 T + 23 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 2 T + 8 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 23 | $D_{4}$ | \( 1 - 8 T + 50 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 4 T + 59 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 2 T + 51 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 18 T + 152 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 12 T + 115 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 2 T + 60 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 14 T + 140 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 6 T + 112 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 8 T + 107 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 16 T + 138 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 8 T + 138 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 6 T + 152 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 24 T + 290 T^{2} - 24 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 2 T + 20 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 151 T^{2} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 10 T + 192 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
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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
−7.918025598287048478910708342389, −7.88844523133213342571001984695, −7.51523540722740531308229735915, −7.30208932756713156689212183245, −6.80833676360941579980853565176, −6.69903899654039129997826940152, −5.84328733303433553313599757849, −5.77932795454930521914528554029, −5.10437856669328546025386083364, −5.01672255524414441883629256536, −4.56586654896427264561337373447, −4.50330550232490266598249739021, −3.56644737974680007462469442914, −3.56577772550821078556392988825, −2.87745437051693930374776064746, −2.67651309792834512550544128024, −2.05799637988956917764669274331, −1.66836285028455817991965921129, −0.857716565104014821301964236767, −0.41008447588596848187422579742,
0.41008447588596848187422579742, 0.857716565104014821301964236767, 1.66836285028455817991965921129, 2.05799637988956917764669274331, 2.67651309792834512550544128024, 2.87745437051693930374776064746, 3.56577772550821078556392988825, 3.56644737974680007462469442914, 4.50330550232490266598249739021, 4.56586654896427264561337373447, 5.01672255524414441883629256536, 5.10437856669328546025386083364, 5.77932795454930521914528554029, 5.84328733303433553313599757849, 6.69903899654039129997826940152, 6.80833676360941579980853565176, 7.30208932756713156689212183245, 7.51523540722740531308229735915, 7.88844523133213342571001984695, 7.918025598287048478910708342389
