| L(s) = 1 | + 2-s − 6·5-s − 8-s − 6·10-s + 6·11-s + 5·13-s − 16-s − 3·17-s + 5·19-s + 6·22-s + 6·23-s + 17·25-s + 5·26-s − 3·29-s − 4·31-s − 3·34-s + 7·37-s + 5·38-s + 6·40-s + 9·41-s − 11·43-s + 6·46-s + 17·50-s − 3·53-s − 36·55-s − 3·58-s − 12·59-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 2.68·5-s − 0.353·8-s − 1.89·10-s + 1.80·11-s + 1.38·13-s − 1/4·16-s − 0.727·17-s + 1.14·19-s + 1.27·22-s + 1.25·23-s + 17/5·25-s + 0.980·26-s − 0.557·29-s − 0.718·31-s − 0.514·34-s + 1.15·37-s + 0.811·38-s + 0.948·40-s + 1.40·41-s − 1.67·43-s + 0.884·46-s + 2.40·50-s − 0.412·53-s − 4.85·55-s − 0.393·58-s − 1.56·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7001316 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7001316 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.822570033\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.822570033\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| good | 5 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 3 T - 8 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 5 T + 6 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 + 3 T - 20 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 - 7 T + 12 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 9 T + 40 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 11 T + 78 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 3 T - 44 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 12 T + 85 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 2 T - 57 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 4 T - 51 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 11 T + 48 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 8 T - 15 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 3 T - 74 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 15 T + 136 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + T - 96 T^{2} + p T^{3} + p^{2} T^{4} \) |
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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
−9.127512573061305029425760939494, −8.466261785404403585366789354703, −8.284724291360690195773354397034, −7.982863134860069762339449149595, −7.45630031383933194224886294360, −7.12289231534284613602805681129, −6.89571198473592990235967409170, −6.30812179500155645160111115047, −6.13622529809761780623611474799, −5.39996888839655529982530980354, −5.07242371475270528210500785488, −4.37091637727000463277538109781, −4.26704044848022488895253474557, −3.76972021160970352689687162476, −3.73500277979377280086059963313, −3.11134340122041873026647462203, −2.85085217907666056857592669651, −1.64391937376548557638798864871, −1.15296223501929745547087845591, −0.46642438152612757018572555655,
0.46642438152612757018572555655, 1.15296223501929745547087845591, 1.64391937376548557638798864871, 2.85085217907666056857592669651, 3.11134340122041873026647462203, 3.73500277979377280086059963313, 3.76972021160970352689687162476, 4.26704044848022488895253474557, 4.37091637727000463277538109781, 5.07242371475270528210500785488, 5.39996888839655529982530980354, 6.13622529809761780623611474799, 6.30812179500155645160111115047, 6.89571198473592990235967409170, 7.12289231534284613602805681129, 7.45630031383933194224886294360, 7.982863134860069762339449149595, 8.284724291360690195773354397034, 8.466261785404403585366789354703, 9.127512573061305029425760939494
