| L(s) = 1 | − 2·2-s − 4-s − 5-s − 3·7-s + 8·8-s + 2·10-s + 5·13-s + 6·14-s − 7·16-s + 3·17-s − 19-s + 20-s − 7·23-s + 5·25-s − 10·26-s + 3·28-s + 3·29-s − 5·31-s − 14·32-s − 6·34-s + 3·35-s + 9·37-s + 2·38-s − 8·40-s + 20·41-s − 8·43-s + 14·46-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 1/2·4-s − 0.447·5-s − 1.13·7-s + 2.82·8-s + 0.632·10-s + 1.38·13-s + 1.60·14-s − 7/4·16-s + 0.727·17-s − 0.229·19-s + 0.223·20-s − 1.45·23-s + 25-s − 1.96·26-s + 0.566·28-s + 0.557·29-s − 0.898·31-s − 2.47·32-s − 1.02·34-s + 0.507·35-s + 1.47·37-s + 0.324·38-s − 1.26·40-s + 3.12·41-s − 1.21·43-s + 2.06·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 149769 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 149769 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3988139624\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3988139624\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 43 | $C_2$ | \( 1 + 8 T + p T^{2} \) |
| good | 2 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 5 | $C_2^2$ | \( 1 + T - 4 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 3 T + 2 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 3 T - 8 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + 7 T + 26 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 3 T - 20 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 5 T - 6 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 9 T + 44 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 + 5 T - 28 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 3 T - 58 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + T - 70 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 11 T + 48 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 5 T - 54 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 9 T - 2 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + T - 88 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
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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
−11.09462155300758319897520107671, −10.93265015643147667652644050703, −10.57056392768403949607736870881, −10.03275525722517290276607768072, −9.573386552916146166544281910704, −9.264506488341810633380950090586, −8.938664893431906517516752534196, −8.416338582271964854707702557315, −7.87025459757384297811958532957, −7.73863488900462569984602735480, −7.10391791558200801367221049154, −6.34740161396081794023446720278, −5.89669917239444553617782279913, −5.34459878062701987007055727537, −4.30385207335211798804657401971, −4.18353268518270748338903154688, −3.58071079634792758129270294535, −2.68155491891568906174088931547, −1.38250738651934570030117821060, −0.61088747845169900649316933416,
0.61088747845169900649316933416, 1.38250738651934570030117821060, 2.68155491891568906174088931547, 3.58071079634792758129270294535, 4.18353268518270748338903154688, 4.30385207335211798804657401971, 5.34459878062701987007055727537, 5.89669917239444553617782279913, 6.34740161396081794023446720278, 7.10391791558200801367221049154, 7.73863488900462569984602735480, 7.87025459757384297811958532957, 8.416338582271964854707702557315, 8.938664893431906517516752534196, 9.264506488341810633380950090586, 9.573386552916146166544281910704, 10.03275525722517290276607768072, 10.57056392768403949607736870881, 10.93265015643147667652644050703, 11.09462155300758319897520107671
