| L(s) = 1 | + 2-s − 2·4-s + 5·7-s − 3·8-s − 9-s − 13-s + 5·14-s + 16-s − 3·17-s − 18-s − 7·19-s − 5·23-s − 26-s − 10·28-s − 29-s − 5·31-s + 2·32-s − 3·34-s + 2·36-s − 4·37-s − 7·38-s + 9·41-s − 8·43-s − 5·46-s − 2·47-s + 6·49-s + 2·52-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 4-s + 1.88·7-s − 1.06·8-s − 1/3·9-s − 0.277·13-s + 1.33·14-s + 1/4·16-s − 0.727·17-s − 0.235·18-s − 1.60·19-s − 1.04·23-s − 0.196·26-s − 1.88·28-s − 0.185·29-s − 0.898·31-s + 0.353·32-s − 0.514·34-s + 1/3·36-s − 0.657·37-s − 1.13·38-s + 1.40·41-s − 1.21·43-s − 0.737·46-s − 0.291·47-s + 6/7·49-s + 0.277·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15405625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15405625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 5 | | \( 1 \) |
| 157 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 2 | $D_{4}$ | \( 1 - T + 3 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 3 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 5 T + 19 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 17 T^{2} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + T + 15 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 3 T + 25 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_4$ | \( 1 + 7 T + 39 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 5 T + 21 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + T + 27 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 5 T + 57 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 4 T + 33 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_4$ | \( 1 - 9 T + 71 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $D_{4}$ | \( 1 + 2 T + 90 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 2 T + 27 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 3 T + p T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 5 T + 67 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 7 T - 5 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 14 T + 171 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 79 | $D_{4}$ | \( 1 + 27 T + 329 T^{2} + 27 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 4 T + 45 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 18 T + 239 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 10 T + 199 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
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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
−8.252641705222639004859054976521, −8.246469348812338411015504805923, −7.53678404318678304645641547398, −7.24438950883203891552135889163, −6.82804186150698664041543725755, −6.32717654571396592162920785619, −5.81406223708427070048555721531, −5.61921415327819098885132660466, −5.14623666969382209105414410835, −4.89036980670697429922057795048, −4.34893621131890048361880924570, −4.24994728989795780556525052714, −3.96719332129002658802599011850, −3.36796248936581007949556717135, −2.65517858123427173923853462658, −2.29795947223272858791009403884, −1.66677778795846286908516252028, −1.38731314152027685207057346991, 0, 0,
1.38731314152027685207057346991, 1.66677778795846286908516252028, 2.29795947223272858791009403884, 2.65517858123427173923853462658, 3.36796248936581007949556717135, 3.96719332129002658802599011850, 4.24994728989795780556525052714, 4.34893621131890048361880924570, 4.89036980670697429922057795048, 5.14623666969382209105414410835, 5.61921415327819098885132660466, 5.81406223708427070048555721531, 6.32717654571396592162920785619, 6.82804186150698664041543725755, 7.24438950883203891552135889163, 7.53678404318678304645641547398, 8.246469348812338411015504805923, 8.252641705222639004859054976521
