| L(s) = 1 | + 2-s − 6·5-s − 4·7-s − 8-s − 6·10-s + 6·11-s − 2·13-s − 4·14-s − 16-s + 6·17-s − 2·19-s + 6·22-s + 12·23-s + 17·25-s − 2·26-s + 9·29-s + 7·31-s + 6·34-s + 24·35-s + 10·37-s − 2·38-s + 6·40-s + 4·43-s + 12·46-s + 12·47-s + 9·49-s + 17·50-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 2.68·5-s − 1.51·7-s − 0.353·8-s − 1.89·10-s + 1.80·11-s − 0.554·13-s − 1.06·14-s − 1/4·16-s + 1.45·17-s − 0.458·19-s + 1.27·22-s + 2.50·23-s + 17/5·25-s − 0.392·26-s + 1.67·29-s + 1.25·31-s + 1.02·34-s + 4.05·35-s + 1.64·37-s − 0.324·38-s + 0.948·40-s + 0.609·43-s + 1.76·46-s + 1.75·47-s + 9/7·49-s + 2.40·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1285956 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1285956 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.690662146\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.690662146\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 6 T + 19 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 2 T - 15 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 9 T + 52 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 4 T - 27 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 12 T + 97 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 3 T - 44 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 3 T - 50 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 4 T - 45 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 2 T - 63 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 2 T - 69 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 5 T - 54 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 9 T - 2 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 6 T - 53 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 13 T + 72 T^{2} - 13 p T^{3} + p^{2} T^{4} \) |
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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
−9.787139695390997440403493160242, −9.726054653558983734764329727628, −9.070680312802161235305342108040, −8.851647777639541928662659320889, −8.353231050658178692623954606532, −7.85397161044719272567085924444, −7.52661575484205842675603170925, −7.06952485579367031972517453728, −6.63261219628726278398161055772, −6.48825635455924513510815005127, −5.81213056561974991646725969432, −5.21324057022628905359156653046, −4.49991948914571717453111671184, −4.24487051295829938467574555157, −4.02488429265996079437928500239, −3.38669133562200595296484428438, −2.96768067327223288966911017286, −2.82458963834384751437891636840, −0.975484638579017762636638776897, −0.70598724177145393018985165130,
0.70598724177145393018985165130, 0.975484638579017762636638776897, 2.82458963834384751437891636840, 2.96768067327223288966911017286, 3.38669133562200595296484428438, 4.02488429265996079437928500239, 4.24487051295829938467574555157, 4.49991948914571717453111671184, 5.21324057022628905359156653046, 5.81213056561974991646725969432, 6.48825635455924513510815005127, 6.63261219628726278398161055772, 7.06952485579367031972517453728, 7.52661575484205842675603170925, 7.85397161044719272567085924444, 8.353231050658178692623954606532, 8.851647777639541928662659320889, 9.070680312802161235305342108040, 9.726054653558983734764329727628, 9.787139695390997440403493160242
