| L(s) = 1 | + 2·2-s + 3·4-s − 2·5-s − 2·7-s + 4·8-s − 4·10-s − 2·11-s − 4·14-s + 5·16-s − 6·20-s − 4·22-s − 4·23-s + 3·25-s − 6·28-s − 4·29-s + 8·31-s + 6·32-s + 4·35-s − 8·40-s − 4·41-s − 6·44-s − 8·46-s + 3·49-s + 6·50-s − 12·53-s + 4·55-s − 8·56-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 3/2·4-s − 0.894·5-s − 0.755·7-s + 1.41·8-s − 1.26·10-s − 0.603·11-s − 1.06·14-s + 5/4·16-s − 1.34·20-s − 0.852·22-s − 0.834·23-s + 3/5·25-s − 1.13·28-s − 0.742·29-s + 1.43·31-s + 1.06·32-s + 0.676·35-s − 1.26·40-s − 0.624·41-s − 0.904·44-s − 1.17·46-s + 3/7·49-s + 0.848·50-s − 1.64·53-s + 0.539·55-s − 1.06·56-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 48024900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48024900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 7 | $C_1$ | \( ( 1 + T )^{2} \) |
| 11 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 13 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 22 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $D_{4}$ | \( 1 + 4 T + 38 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 4 T + 14 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 8 T + 30 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 46 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 61 | $D_{4}$ | \( 1 - 4 T + 78 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 4 T + 126 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 16 T + 158 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 4 T - 42 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 24 T + 310 T^{2} + 24 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
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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
−7.67748903489183673708373646158, −7.48182660990078748891563576285, −6.83037374498026632908238186793, −6.79937550075434740879568938379, −6.18747179445599159935963859434, −6.15030972567659718751926779198, −5.54986991649816838480846130667, −5.36170187812529085077880780259, −4.77002518917864082764774613405, −4.58460660720671086609781968673, −4.02439405640035891510239979434, −4.00815799229655652624240862658, −3.29323239198889336831080552838, −3.15013242564107021027956354890, −2.60092616688986602321141820903, −2.47555445931060146533630769858, −1.43554775972807461350750042505, −1.39740166594518336368911085882, 0, 0,
1.39740166594518336368911085882, 1.43554775972807461350750042505, 2.47555445931060146533630769858, 2.60092616688986602321141820903, 3.15013242564107021027956354890, 3.29323239198889336831080552838, 4.00815799229655652624240862658, 4.02439405640035891510239979434, 4.58460660720671086609781968673, 4.77002518917864082764774613405, 5.36170187812529085077880780259, 5.54986991649816838480846130667, 6.15030972567659718751926779198, 6.18747179445599159935963859434, 6.79937550075434740879568938379, 6.83037374498026632908238186793, 7.48182660990078748891563576285, 7.67748903489183673708373646158
