| L(s) = 1 | − 2-s + 3-s + 4-s − 6-s − 8-s − 2·9-s − 2·11-s + 12-s + 4·13-s + 16-s + 2·18-s + 2·22-s − 3·23-s − 24-s − 4·25-s − 4·26-s − 5·27-s − 32-s − 2·33-s − 2·36-s + 7·37-s + 4·39-s − 2·44-s + 3·46-s − 15·47-s + 48-s − 13·49-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s − 0.353·8-s − 2/3·9-s − 0.603·11-s + 0.288·12-s + 1.10·13-s + 1/4·16-s + 0.471·18-s + 0.426·22-s − 0.625·23-s − 0.204·24-s − 4/5·25-s − 0.784·26-s − 0.962·27-s − 0.176·32-s − 0.348·33-s − 1/3·36-s + 1.15·37-s + 0.640·39-s − 0.301·44-s + 0.442·46-s − 2.18·47-s + 0.144·48-s − 1.85·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 224928 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224928 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 + T \) |
| 3 | $C_2$ | \( 1 - T + p T^{2} \) |
| 11 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 3 T + p T^{2} ) \) |
| 71 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 12 T + p T^{2} ) \) |
| good | 5 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 22 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 16 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 - 41 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 56 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 - 44 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + 115 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 + 67 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 89 | $C_2^2$ | \( 1 - 83 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
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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
−8.760893773925095496051695586562, −8.197915469585743418799648110254, −7.911491112694402245470444100870, −7.78761560243761427098468502816, −6.86995494636192185776861216733, −6.39594282138785523026627045038, −5.93922161398863369478060303879, −5.53287351212453581595986823504, −4.74292780666677034166823878914, −4.11234316512471641160747478728, −3.30065640491588331407175638844, −3.03528723819405541952760377190, −2.14839230414015339524515185015, −1.47790914187423188241944718100, 0,
1.47790914187423188241944718100, 2.14839230414015339524515185015, 3.03528723819405541952760377190, 3.30065640491588331407175638844, 4.11234316512471641160747478728, 4.74292780666677034166823878914, 5.53287351212453581595986823504, 5.93922161398863369478060303879, 6.39594282138785523026627045038, 6.86995494636192185776861216733, 7.78761560243761427098468502816, 7.911491112694402245470444100870, 8.197915469585743418799648110254, 8.760893773925095496051695586562
