| L(s) = 1 | − 2-s + 4-s − 8-s − 8·11-s + 16-s − 12·19-s + 8·22-s − 6·25-s − 32-s + 12·38-s + 12·41-s − 16·43-s − 8·44-s − 10·49-s + 6·50-s + 8·59-s + 64-s − 4·67-s + 28·73-s − 12·76-s − 12·82-s − 24·83-s + 16·86-s + 8·88-s − 12·89-s − 20·97-s + 10·98-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.353·8-s − 2.41·11-s + 1/4·16-s − 2.75·19-s + 1.70·22-s − 6/5·25-s − 0.176·32-s + 1.94·38-s + 1.87·41-s − 2.43·43-s − 1.20·44-s − 1.42·49-s + 0.848·50-s + 1.04·59-s + 1/8·64-s − 0.488·67-s + 3.27·73-s − 1.37·76-s − 1.32·82-s − 2.63·83-s + 1.72·86-s + 0.852·88-s − 1.27·89-s − 2.03·97-s + 1.01·98-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1752192 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1752192 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( 1 + T \) |
| 3 | | \( 1 \) |
| 13 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 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
−7.76516545555292037952264969735, −6.93707785293490008685942511031, −6.54658570849644716266634653386, −6.31902631650893558385157451977, −5.59584593532556316161391417712, −5.29958688295089777719730657827, −4.89846206181494562517735323880, −4.04504924809123623904528509602, −3.99394475360161838020150844401, −2.94012618851587332451134020886, −2.57523735650192716532994366231, −2.16841436964911272483029187855, −1.50688699187961671245243796764, 0, 0,
1.50688699187961671245243796764, 2.16841436964911272483029187855, 2.57523735650192716532994366231, 2.94012618851587332451134020886, 3.99394475360161838020150844401, 4.04504924809123623904528509602, 4.89846206181494562517735323880, 5.29958688295089777719730657827, 5.59584593532556316161391417712, 6.31902631650893558385157451977, 6.54658570849644716266634653386, 6.93707785293490008685942511031, 7.76516545555292037952264969735
