| L(s) = 1 | − 3·2-s + 3-s + 4·4-s − 3·6-s − 3·7-s − 3·8-s + 6·11-s + 4·12-s − 2·13-s + 9·14-s + 3·16-s − 6·17-s + 3·19-s − 3·21-s − 18·22-s + 6·23-s − 3·24-s + 6·26-s − 27-s − 12·28-s + 6·29-s − 6·32-s + 6·33-s + 18·34-s − 12·37-s − 9·38-s − 2·39-s + ⋯ |
| L(s) = 1 | − 2.12·2-s + 0.577·3-s + 2·4-s − 1.22·6-s − 1.13·7-s − 1.06·8-s + 1.80·11-s + 1.15·12-s − 0.554·13-s + 2.40·14-s + 3/4·16-s − 1.45·17-s + 0.688·19-s − 0.654·21-s − 3.83·22-s + 1.25·23-s − 0.612·24-s + 1.17·26-s − 0.192·27-s − 2.26·28-s + 1.11·29-s − 1.06·32-s + 1.04·33-s + 3.08·34-s − 1.97·37-s − 1.45·38-s − 0.320·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 950625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5229078005\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5229078005\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + T^{2} \) |
| 5 | | \( 1 \) |
| 13 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 + 3 T + 5 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 6 T + 23 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 6 T + 19 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 3 T + 22 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 6 T + 13 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 6 T + 7 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 6 T + 53 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 18 T + 167 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 2 T - 57 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 15 T + 142 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 6 T + 83 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 71 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 11 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 154 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 6 T + 101 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 12 T + 145 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
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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
−10.17409844113043149648880155587, −9.562059119991505303866464557720, −9.316170247193884661939883313119, −8.994557906061185019145934888342, −8.677803451843212725388676751257, −8.429303537140510877994583078199, −8.055414159428132898465799037820, −7.06851153087279721182464193102, −7.02243857894782823215289861566, −6.77872360298652598426971635508, −6.39494726996509608743847341045, −5.36676025542840266162861278458, −5.26267968133969153147753243514, −4.19237411616698096483538981978, −3.85830880330214407668358632296, −3.23840063541399958482986344107, −2.70893890258472004785198168838, −1.98649813611342951146154698773, −1.23361121936444561154239974652, −0.51977396405764375520712673252,
0.51977396405764375520712673252, 1.23361121936444561154239974652, 1.98649813611342951146154698773, 2.70893890258472004785198168838, 3.23840063541399958482986344107, 3.85830880330214407668358632296, 4.19237411616698096483538981978, 5.26267968133969153147753243514, 5.36676025542840266162861278458, 6.39494726996509608743847341045, 6.77872360298652598426971635508, 7.02243857894782823215289861566, 7.06851153087279721182464193102, 8.055414159428132898465799037820, 8.429303537140510877994583078199, 8.677803451843212725388676751257, 8.994557906061185019145934888342, 9.316170247193884661939883313119, 9.562059119991505303866464557720, 10.17409844113043149648880155587
