L(s) = 1 | − 9-s + 8·19-s + 6·29-s − 14·31-s − 18·41-s + 13·49-s − 6·59-s − 20·61-s + 18·71-s − 16·79-s + 81-s − 6·101-s + 8·109-s − 22·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 10·169-s − 8·171-s + 173-s + 179-s + ⋯ |
L(s) = 1 | − 1/3·9-s + 1.83·19-s + 1.11·29-s − 2.51·31-s − 2.81·41-s + 13/7·49-s − 0.781·59-s − 2.56·61-s + 2.13·71-s − 1.80·79-s + 1/9·81-s − 0.597·101-s + 0.766·109-s − 2·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 0.769·169-s − 0.611·171-s + 0.0760·173-s + 0.0747·179-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 47610000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 47610000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.595173103\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.595173103\) |
\(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_2$ | \( 1 + T^{2} \) |
| 5 | | \( 1 \) |
| 23 | $C_2$ | \( 1 + T^{2} \) |
good | 7 | $C_2^2$ | \( 1 - 13 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 47 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 35 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 59 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 190 T^{2} + p^{2} T^{4} \) |
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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
−8.040137664935669033519495330327, −7.78292729899159681713615993103, −7.31086571659068924773226946475, −7.20947023688811931026935980631, −6.86744077953363933208801249477, −6.26725986814166630022437696395, −6.12519237821493111299049909418, −5.55711147808697817417414054391, −5.19466808682169132593165654582, −5.18847283552332095693484090054, −4.61779025839317162789063680333, −4.14218874847360148194961915881, −3.61413455421682920703527380155, −3.46039571636784232692857879691, −2.89063515307166714863420891091, −2.72775880597307823600690001192, −1.84229464301911745175299685249, −1.68030843271921561787969637683, −1.05126687710298547130224223050, −0.33292646355384807422317182810,
0.33292646355384807422317182810, 1.05126687710298547130224223050, 1.68030843271921561787969637683, 1.84229464301911745175299685249, 2.72775880597307823600690001192, 2.89063515307166714863420891091, 3.46039571636784232692857879691, 3.61413455421682920703527380155, 4.14218874847360148194961915881, 4.61779025839317162789063680333, 5.18847283552332095693484090054, 5.19466808682169132593165654582, 5.55711147808697817417414054391, 6.12519237821493111299049909418, 6.26725986814166630022437696395, 6.86744077953363933208801249477, 7.20947023688811931026935980631, 7.31086571659068924773226946475, 7.78292729899159681713615993103, 8.040137664935669033519495330327