| L(s) = 1 | + 2-s + 2·3-s + 4-s + 2·5-s + 2·6-s + 2·7-s + 3·8-s + 3·9-s + 2·10-s + 2·12-s + 7·13-s + 2·14-s + 4·15-s + 16-s + 3·18-s + 12·19-s + 2·20-s + 4·21-s − 8·23-s + 6·24-s − 7·25-s + 7·26-s + 4·27-s + 2·28-s + 29-s + 4·30-s − 32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.15·3-s + 1/2·4-s + 0.894·5-s + 0.816·6-s + 0.755·7-s + 1.06·8-s + 9-s + 0.632·10-s + 0.577·12-s + 1.94·13-s + 0.534·14-s + 1.03·15-s + 1/4·16-s + 0.707·18-s + 2.75·19-s + 0.447·20-s + 0.872·21-s − 1.66·23-s + 1.22·24-s − 7/5·25-s + 1.37·26-s + 0.769·27-s + 0.377·28-s + 0.185·29-s + 0.730·30-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6456681 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6456681 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(12.31205784\) |
| \(L(\frac12)\) |
\(\approx\) |
\(12.31205784\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
| 11 | | \( 1 \) |
| good | 2 | $D_{4}$ | \( 1 - T - p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 13 | $D_{4}$ | \( 1 - 7 T + 34 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 29 | $D_{4}$ | \( 1 - T + 20 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 7 T + 48 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 11 T + 108 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - T + 48 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 11 T + 120 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 5 T + 6 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 21 T + 224 T^{2} + 21 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 2 T - 30 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 3 T + 30 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 2 T + 126 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 6 T + 138 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 2 T + 142 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 3 T + 62 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 6 T + 119 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 15 T + 212 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
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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
−8.965099917675267783972131160171, −8.808791178186706025377966303338, −8.193320392382910427986031826087, −7.83752791360058466545958936186, −7.68263298357689888953103921350, −7.37263028310462705418306958853, −6.82315556932055083436163382516, −6.18254409704501890100122302142, −5.92331233632029478084409522523, −5.77517917215845138492114095406, −4.98499229255871837576969402770, −4.86137357131977610410945099514, −4.07219718590699500777615701967, −3.96545678245895010800091965970, −3.40038788002161346957370063503, −3.04336360006953761694513366437, −2.38433419671920514774134749769, −1.70840803898572090112120367954, −1.67473636080385829243454923709, −1.03934814603548413725743981788,
1.03934814603548413725743981788, 1.67473636080385829243454923709, 1.70840803898572090112120367954, 2.38433419671920514774134749769, 3.04336360006953761694513366437, 3.40038788002161346957370063503, 3.96545678245895010800091965970, 4.07219718590699500777615701967, 4.86137357131977610410945099514, 4.98499229255871837576969402770, 5.77517917215845138492114095406, 5.92331233632029478084409522523, 6.18254409704501890100122302142, 6.82315556932055083436163382516, 7.37263028310462705418306958853, 7.68263298357689888953103921350, 7.83752791360058466545958936186, 8.193320392382910427986031826087, 8.808791178186706025377966303338, 8.965099917675267783972131160171
