| L(s) = 1 | − 3·2-s − 3·3-s + 4·4-s − 6·5-s + 9·6-s − 5·7-s − 3·8-s + 6·9-s + 18·10-s − 12·12-s − 3·13-s + 15·14-s + 18·15-s + 3·16-s + 3·17-s − 18·18-s − 9·19-s − 24·20-s + 15·21-s + 9·24-s + 17·25-s + 9·26-s − 9·27-s − 20·28-s − 9·29-s − 54·30-s − 6·31-s + ⋯ |
| L(s) = 1 | − 2.12·2-s − 1.73·3-s + 2·4-s − 2.68·5-s + 3.67·6-s − 1.88·7-s − 1.06·8-s + 2·9-s + 5.69·10-s − 3.46·12-s − 0.832·13-s + 4.00·14-s + 4.64·15-s + 3/4·16-s + 0.727·17-s − 4.24·18-s − 2.06·19-s − 5.36·20-s + 3.27·21-s + 1.83·24-s + 17/5·25-s + 1.76·26-s − 1.73·27-s − 3.77·28-s − 1.67·29-s − 9.85·30-s − 1.07·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 3 | $C_2$ | \( 1 + p T + p T^{2} \) |
| 7 | $C_2$ | \( 1 + 5 T + p T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 + 3 T + 5 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 19 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 3 T - 8 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 19 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 9 T + 56 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 6 T + 43 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 7 T + 12 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 3 T - 32 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + T - 42 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 15 T + 128 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 24 T + 253 T^{2} - 24 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 4 T - 51 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 9 T + 100 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 8 T - 15 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 15 T + 142 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 3 T - 80 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 3 T + 100 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
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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
−15.06020113073052208804577183466, −14.72518720460067095407821569932, −12.97195490744740051471590904968, −12.63168405721202051569981545816, −12.33391187919379767374474263630, −11.74329413950311306917841786480, −10.96934280226174371830932762481, −10.92088681910152773336161361167, −9.864422386167622675789850927708, −9.835219782869175814349722782671, −8.802172192593933841385547815591, −8.288463075388603889105459389983, −7.34549107508382060348600072128, −7.28822556449389205247566826803, −6.47914789526070544924075624035, −5.52642751154064048615476871412, −4.19208744532054610463242934762, −3.59247465438609684224541002929, 0, 0,
3.59247465438609684224541002929, 4.19208744532054610463242934762, 5.52642751154064048615476871412, 6.47914789526070544924075624035, 7.28822556449389205247566826803, 7.34549107508382060348600072128, 8.288463075388603889105459389983, 8.802172192593933841385547815591, 9.835219782869175814349722782671, 9.864422386167622675789850927708, 10.92088681910152773336161361167, 10.96934280226174371830932762481, 11.74329413950311306917841786480, 12.33391187919379767374474263630, 12.63168405721202051569981545816, 12.97195490744740051471590904968, 14.72518720460067095407821569932, 15.06020113073052208804577183466
