| L(s) = 1 | − 4-s + 2·9-s − 8·11-s + 16-s + 12·19-s + 12·29-s + 8·31-s − 2·36-s + 8·41-s + 8·44-s + 28·59-s − 20·61-s − 64-s + 24·71-s − 12·76-s − 8·79-s − 5·81-s − 16·89-s − 16·99-s − 4·101-s − 20·109-s − 12·116-s + 26·121-s − 8·124-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | − 1/2·4-s + 2/3·9-s − 2.41·11-s + 1/4·16-s + 2.75·19-s + 2.22·29-s + 1.43·31-s − 1/3·36-s + 1.24·41-s + 1.20·44-s + 3.64·59-s − 2.56·61-s − 1/8·64-s + 2.84·71-s − 1.37·76-s − 0.900·79-s − 5/9·81-s − 1.69·89-s − 1.60·99-s − 0.398·101-s − 1.91·109-s − 1.11·116-s + 2.36·121-s − 0.718·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6002500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6002500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.521717454\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.521717454\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 5 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 78 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 118 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 162 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
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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
−9.309715555082081133316007707851, −8.476377517317268353250078622555, −8.381995041853395292428856848994, −8.014243131926179487228188988593, −7.60256676112824481143366506774, −7.42266567160512747634215248184, −6.72428270293329093466002835358, −6.68470421017370249373208073045, −5.80978254493715340456863322811, −5.39208242529338612975099560943, −5.34131717998721515322648589431, −4.86952389118626788525970680761, −4.30483223082779562216024553941, −4.13077031631429424517547442500, −3.05145050008736289448953964345, −3.04981496558464750891819275428, −2.66175036468782064581884762787, −1.89360482654276718316681218117, −0.924637238044766700621084998264, −0.72364501131582955918842644560,
0.72364501131582955918842644560, 0.924637238044766700621084998264, 1.89360482654276718316681218117, 2.66175036468782064581884762787, 3.04981496558464750891819275428, 3.05145050008736289448953964345, 4.13077031631429424517547442500, 4.30483223082779562216024553941, 4.86952389118626788525970680761, 5.34131717998721515322648589431, 5.39208242529338612975099560943, 5.80978254493715340456863322811, 6.68470421017370249373208073045, 6.72428270293329093466002835358, 7.42266567160512747634215248184, 7.60256676112824481143366506774, 8.014243131926179487228188988593, 8.381995041853395292428856848994, 8.476377517317268353250078622555, 9.309715555082081133316007707851
