| L(s) = 1 | + 2-s − 5·7-s − 8-s + 4·11-s + 7·13-s − 5·14-s − 16-s + 4·17-s + 2·19-s + 4·22-s − 4·23-s + 5·25-s + 7·26-s + 6·29-s + 4·34-s + 11·37-s + 2·38-s − 2·41-s + 43-s − 4·46-s + 18·49-s + 5·50-s + 4·53-s + 5·56-s + 6·58-s + 4·59-s + 2·61-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.88·7-s − 0.353·8-s + 1.20·11-s + 1.94·13-s − 1.33·14-s − 1/4·16-s + 0.970·17-s + 0.458·19-s + 0.852·22-s − 0.834·23-s + 25-s + 1.37·26-s + 1.11·29-s + 0.685·34-s + 1.80·37-s + 0.324·38-s − 0.312·41-s + 0.152·43-s − 0.589·46-s + 18/7·49-s + 0.707·50-s + 0.549·53-s + 0.668·56-s + 0.787·58-s + 0.520·59-s + 0.256·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2683044 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2683044 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.724575577\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.724575577\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + T^{2} \) |
| 3 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + 5 T + p T^{2} \) |
| 13 | $C_2$ | \( 1 - 7 T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 4 T - T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 4 T - 7 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 6 T + 7 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 2 T - 37 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - T - 42 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 4 T - 37 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 4 T - 43 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 6 T - 35 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 + 8 T - 15 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 6 T - 53 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 9 T - 16 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
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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
−9.492552658617450015680956644991, −9.287229556594322777789153513250, −8.763451934779689418526030583315, −8.599183970891564117345151836622, −7.83286140191494661844101463086, −7.76419947217800253939898949969, −6.79313672186367467971776428689, −6.56118724226602761641323561495, −6.44401293987222095852165470873, −6.05582015850600244282979399414, −5.52533661375818056346823228281, −5.18344274712291435566187268629, −4.45346536707687291745801506125, −3.87075675872381430613277272482, −3.65146432815264680377760561571, −3.45481222684985650914265716600, −2.77674277980236485211892353745, −2.25543731207904531159590457161, −0.953602280718554775328669010184, −0.949156937575896481977585456579,
0.949156937575896481977585456579, 0.953602280718554775328669010184, 2.25543731207904531159590457161, 2.77674277980236485211892353745, 3.45481222684985650914265716600, 3.65146432815264680377760561571, 3.87075675872381430613277272482, 4.45346536707687291745801506125, 5.18344274712291435566187268629, 5.52533661375818056346823228281, 6.05582015850600244282979399414, 6.44401293987222095852165470873, 6.56118724226602761641323561495, 6.79313672186367467971776428689, 7.76419947217800253939898949969, 7.83286140191494661844101463086, 8.599183970891564117345151836622, 8.763451934779689418526030583315, 9.287229556594322777789153513250, 9.492552658617450015680956644991
