| L(s) = 1 | − 2·2-s − 3·3-s + 3·4-s + 6·5-s + 6·6-s − 7-s − 4·8-s + 6·9-s − 12·10-s + 6·11-s − 9·12-s − 5·13-s + 2·14-s − 18·15-s + 5·16-s − 12·18-s + 5·19-s + 18·20-s + 3·21-s − 12·22-s + 12·24-s + 19·25-s + 10·26-s − 9·27-s − 3·28-s + 12·29-s + 36·30-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 1.73·3-s + 3/2·4-s + 2.68·5-s + 2.44·6-s − 0.377·7-s − 1.41·8-s + 2·9-s − 3.79·10-s + 1.80·11-s − 2.59·12-s − 1.38·13-s + 0.534·14-s − 4.64·15-s + 5/4·16-s − 2.82·18-s + 1.14·19-s + 4.02·20-s + 0.654·21-s − 2.55·22-s + 2.44·24-s + 19/5·25-s + 1.96·26-s − 1.73·27-s − 0.566·28-s + 2.22·29-s + 6.57·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 298116 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 298116 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.019683270\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.019683270\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 3 | $C_2$ | \( 1 + p T + p T^{2} \) |
| 7 | $C_2$ | \( 1 + T + p T^{2} \) |
| 13 | $C_2$ | \( 1 + 5 T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - 6 T + 17 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 6 T + 25 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 5 T + 6 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 12 T + 77 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 8 T + 33 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 18 T + 149 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + T - 42 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 12 T + 95 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 6 T + 65 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 - 9 T + 88 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 8 T - 15 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 154 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 166 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 7 T - 48 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
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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
−10.73378141141859610697875123829, −10.56230978220468315845599432778, −9.875339453351827224945861881078, −9.843691173542517022588394941901, −9.362358875780042720200944838944, −9.255990877082562062631749253532, −8.712119187393681288506658530560, −7.77214652439216223516170469366, −7.06738395159695094126951902046, −7.00967729811912764498299557089, −6.33561102252767572507756785780, −6.16723364235994099169436458262, −5.69357377689222093803945954762, −5.29053477758604399199797271050, −4.72821050387656558243373913926, −3.85227911136320157557543189351, −2.63357097597375079113730642612, −2.25939069747099897410742993062, −1.29881398216898491119744554424, −0.977735717229104235874247603134,
0.977735717229104235874247603134, 1.29881398216898491119744554424, 2.25939069747099897410742993062, 2.63357097597375079113730642612, 3.85227911136320157557543189351, 4.72821050387656558243373913926, 5.29053477758604399199797271050, 5.69357377689222093803945954762, 6.16723364235994099169436458262, 6.33561102252767572507756785780, 7.00967729811912764498299557089, 7.06738395159695094126951902046, 7.77214652439216223516170469366, 8.712119187393681288506658530560, 9.255990877082562062631749253532, 9.362358875780042720200944838944, 9.843691173542517022588394941901, 9.875339453351827224945861881078, 10.56230978220468315845599432778, 10.73378141141859610697875123829
