| L(s) = 1 | − 2-s − 5-s + 7-s + 8-s + 10-s − 5·11-s − 14-s − 16-s + 4·17-s − 2·19-s + 5·22-s + 23-s + 5·25-s − 4·29-s + 9·31-s − 4·34-s − 35-s + 10·37-s + 2·38-s − 40-s + 9·41-s + 10·43-s − 46-s − 6·47-s − 5·50-s + 24·53-s + 5·55-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.447·5-s + 0.377·7-s + 0.353·8-s + 0.316·10-s − 1.50·11-s − 0.267·14-s − 1/4·16-s + 0.970·17-s − 0.458·19-s + 1.06·22-s + 0.208·23-s + 25-s − 0.742·29-s + 1.61·31-s − 0.685·34-s − 0.169·35-s + 1.64·37-s + 0.324·38-s − 0.158·40-s + 1.40·41-s + 1.52·43-s − 0.147·46-s − 0.875·47-s − 0.707·50-s + 3.29·53-s + 0.674·55-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
| \(L(1)\) |
\(\approx\) |
\(1.222858272\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.222858272\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + T - 4 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 5 T + 14 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - T - 22 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 4 T - 13 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 9 T + 50 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 9 T + 40 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 10 T + 57 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 6 T - 11 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 14 T + 137 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 8 T - 3 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 13 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 6 T - 43 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 4 T - 67 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 16 T + 159 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
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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
−10.09249153676200250276119746691, −9.722551025457789726846914988637, −9.027126650352188801344842607865, −8.820431277888768831931558527740, −8.307770088887986882928349069310, −8.086678215711343290644210372615, −7.46983869278417207048881754803, −7.44772717243905492103760467807, −6.94638715705133832647102307968, −6.17424350685833532434394312595, −5.75921637312770979383939824971, −5.40594257258444688788536376082, −4.86724183069534869203422497547, −4.19364425682168426822832850646, −4.16375786701929451733323131838, −3.15815482586006477424790153876, −2.61764318624072097055462757229, −2.32704008276974662909946581123, −1.15265776637237791357158142894, −0.64691438952779580472167653723,
0.64691438952779580472167653723, 1.15265776637237791357158142894, 2.32704008276974662909946581123, 2.61764318624072097055462757229, 3.15815482586006477424790153876, 4.16375786701929451733323131838, 4.19364425682168426822832850646, 4.86724183069534869203422497547, 5.40594257258444688788536376082, 5.75921637312770979383939824971, 6.17424350685833532434394312595, 6.94638715705133832647102307968, 7.44772717243905492103760467807, 7.46983869278417207048881754803, 8.086678215711343290644210372615, 8.307770088887986882928349069310, 8.820431277888768831931558527740, 9.027126650352188801344842607865, 9.722551025457789726846914988637, 10.09249153676200250276119746691
