| L(s) = 1 | − 3·3-s + 6·9-s + 6·11-s − 4·13-s − 12·23-s − 9·27-s − 18·33-s + 16·37-s + 12·39-s + 12·47-s + 14·49-s + 24·59-s + 16·61-s + 36·69-s − 12·71-s + 2·73-s + 9·81-s + 18·83-s + 20·97-s + 36·99-s − 6·107-s − 16·109-s − 48·111-s − 24·117-s + 5·121-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | − 1.73·3-s + 2·9-s + 1.80·11-s − 1.10·13-s − 2.50·23-s − 1.73·27-s − 3.13·33-s + 2.63·37-s + 1.92·39-s + 1.75·47-s + 2·49-s + 3.12·59-s + 2.04·61-s + 4.33·69-s − 1.42·71-s + 0.234·73-s + 81-s + 1.97·83-s + 2.03·97-s + 3.61·99-s − 0.580·107-s − 1.53·109-s − 4.55·111-s − 2.21·117-s + 5/11·121-s + 0.0887·127-s + 0.0873·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.199484073\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.199484073\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + p T + p T^{2} \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 7 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 55 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 74 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 110 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 151 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
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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
−9.976481671854491664169160539553, −9.754337560460811700069512717332, −9.181192802291979827371125520145, −8.966429718695992216810964432667, −8.120765976651224914736317122282, −7.940277479717745619180806268176, −7.13617777727614912949234822791, −7.11522854541445265490842928416, −6.55410115577573692881082713121, −6.11598497518027464190669059566, −5.69301132670595149418031678926, −5.60239882650989773031994577036, −4.76888859816786094262034067345, −4.37642736800182503572167199721, −3.86356699939317211641291694422, −3.82899820698980028168083646009, −2.30083713219875881569122642647, −2.29441613119590636210046084071, −1.10690793654110656542402701433, −0.63597144735035765118299971050,
0.63597144735035765118299971050, 1.10690793654110656542402701433, 2.29441613119590636210046084071, 2.30083713219875881569122642647, 3.82899820698980028168083646009, 3.86356699939317211641291694422, 4.37642736800182503572167199721, 4.76888859816786094262034067345, 5.60239882650989773031994577036, 5.69301132670595149418031678926, 6.11598497518027464190669059566, 6.55410115577573692881082713121, 7.11522854541445265490842928416, 7.13617777727614912949234822791, 7.940277479717745619180806268176, 8.120765976651224914736317122282, 8.966429718695992216810964432667, 9.181192802291979827371125520145, 9.754337560460811700069512717332, 9.976481671854491664169160539553
