| L(s) = 1 | − 3·2-s + 4·4-s − 3·8-s − 9-s − 6·11-s + 2·13-s + 3·16-s + 6·17-s + 3·18-s − 6·19-s + 18·22-s − 12·23-s − 5·25-s − 6·26-s − 10·31-s − 6·32-s − 18·34-s − 4·36-s − 4·37-s + 18·38-s − 16·43-s − 24·44-s + 36·46-s + 6·47-s + 15·50-s + 8·52-s − 6·53-s + ⋯ |
| L(s) = 1 | − 2.12·2-s + 2·4-s − 1.06·8-s − 1/3·9-s − 1.80·11-s + 0.554·13-s + 3/4·16-s + 1.45·17-s + 0.707·18-s − 1.37·19-s + 3.83·22-s − 2.50·23-s − 25-s − 1.17·26-s − 1.79·31-s − 1.06·32-s − 3.08·34-s − 2/3·36-s − 0.657·37-s + 2.91·38-s − 2.43·43-s − 3.61·44-s + 5.30·46-s + 0.875·47-s + 2.12·50-s + 1.10·52-s − 0.824·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 405769 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405769 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 7 | | \( 1 \) |
| 13 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 2 | $C_2^2$ | \( 1 + 3 T + 5 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 3 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 17 | $D_{4}$ | \( 1 - 6 T + 23 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 23 | $D_{4}$ | \( 1 + 12 T + 77 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 37 | $D_{4}$ | \( 1 + 4 T + 33 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 47 | $D_{4}$ | \( 1 - 6 T + 83 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 6 T + 95 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 6 T + 107 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 8 T + 117 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 8 T + 129 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 173 T^{2} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 8 T + 30 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
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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
−10.26813337319332781669343398813, −9.926967043435727606751488411465, −9.535584811349761561598349820299, −9.073591398621732359811111073913, −8.460241464774015638621155340295, −8.279674204242944344175756530563, −7.979585550272571007696206837437, −7.53959975314164666285963391409, −7.27230104557089325010484293257, −6.20773811119268649499049361066, −6.07649062075562653452378841332, −5.48987176006989565595941322782, −5.02887787297504592985778238416, −4.09084762809047939404241567772, −3.55744827318528851282072200960, −2.90462871799594758035438569900, −1.95910020964794718294263088171, −1.63436554889716791348719925216, 0, 0,
1.63436554889716791348719925216, 1.95910020964794718294263088171, 2.90462871799594758035438569900, 3.55744827318528851282072200960, 4.09084762809047939404241567772, 5.02887787297504592985778238416, 5.48987176006989565595941322782, 6.07649062075562653452378841332, 6.20773811119268649499049361066, 7.27230104557089325010484293257, 7.53959975314164666285963391409, 7.979585550272571007696206837437, 8.279674204242944344175756530563, 8.460241464774015638621155340295, 9.073591398621732359811111073913, 9.535584811349761561598349820299, 9.926967043435727606751488411465, 10.26813337319332781669343398813
