| L(s) = 1 | + 2·2-s + 5·5-s + 4·7-s − 8·8-s + 10·10-s + 12·11-s + 58·13-s + 8·14-s − 16·16-s − 132·17-s − 200·19-s + 24·22-s + 132·23-s + 116·26-s − 90·29-s − 152·31-s − 264·34-s + 20·35-s − 68·37-s − 400·38-s − 40·40-s − 438·41-s − 32·43-s + 264·46-s − 204·47-s + 343·49-s − 444·53-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.447·5-s + 0.215·7-s − 0.353·8-s + 0.316·10-s + 0.328·11-s + 1.23·13-s + 0.152·14-s − 1/4·16-s − 1.88·17-s − 2.41·19-s + 0.232·22-s + 1.19·23-s + 0.874·26-s − 0.576·29-s − 0.880·31-s − 1.33·34-s + 0.0965·35-s − 0.302·37-s − 1.70·38-s − 0.158·40-s − 1.66·41-s − 0.113·43-s + 0.846·46-s − 0.633·47-s + 49-s − 1.15·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 656100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 656100 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.554815638\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.554815638\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2$ | \( 1 - p T + p^{2} T^{2} \) |
| 3 | | \( 1 \) |
| 5 | $C_2$ | \( 1 - p T + p^{2} T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 - 4 T - 327 T^{2} - 4 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 12 T - 1187 T^{2} - 12 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 58 T + 1167 T^{2} - 58 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 66 T + p^{3} T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 100 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 132 T + 5257 T^{2} - 132 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 90 T - 16289 T^{2} + 90 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 152 T - 6687 T^{2} + 152 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 34 T + p^{3} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 438 T + 122923 T^{2} + 438 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 32 T - 78483 T^{2} + 32 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 204 T - 62207 T^{2} + 204 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 222 T + p^{3} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 420 T - 28979 T^{2} - 420 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 902 T + 586623 T^{2} + 902 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 1024 T + 747813 T^{2} - 1024 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 432 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 362 T + p^{3} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 160 T - 467439 T^{2} - 160 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 72 T - 566603 T^{2} - 72 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 810 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 1106 T + 310563 T^{2} + 1106 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−10.24810430484240464237829744159, −9.504340220351672998132276595240, −9.155558070442281733184108638277, −8.711898775279448489341358361673, −8.535954359631225720758120164837, −8.172412969522017232258154424151, −7.29442299112644107098340165881, −6.83359663725355919200559533509, −6.49261622379655524677944765765, −6.24815134221021108176032671924, −5.61929477209712437878053246170, −5.18188610301097466197225727493, −4.45661788166544685773343518797, −4.39440334422058245109952907267, −3.68567570352317426149949588926, −3.27588430535074214534442888341, −2.40142022062648563028881384523, −1.94670932595295927389439150211, −1.42134420234493268272021038845, −0.28173481907848503254773550097,
0.28173481907848503254773550097, 1.42134420234493268272021038845, 1.94670932595295927389439150211, 2.40142022062648563028881384523, 3.27588430535074214534442888341, 3.68567570352317426149949588926, 4.39440334422058245109952907267, 4.45661788166544685773343518797, 5.18188610301097466197225727493, 5.61929477209712437878053246170, 6.24815134221021108176032671924, 6.49261622379655524677944765765, 6.83359663725355919200559533509, 7.29442299112644107098340165881, 8.172412969522017232258154424151, 8.535954359631225720758120164837, 8.711898775279448489341358361673, 9.155558070442281733184108638277, 9.504340220351672998132276595240, 10.24810430484240464237829744159
