| L(s) = 1 | + 2·2-s + 4·3-s + 2·4-s − 2·5-s + 8·6-s + 2·7-s + 8·9-s − 4·10-s + 8·12-s − 8·13-s + 4·14-s − 8·15-s − 4·16-s + 4·17-s + 16·18-s − 4·19-s − 4·20-s + 8·21-s + 8·23-s − 25-s − 16·26-s + 12·27-s + 4·28-s − 16·30-s − 8·32-s + 8·34-s − 4·35-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 2.30·3-s + 4-s − 0.894·5-s + 3.26·6-s + 0.755·7-s + 8/3·9-s − 1.26·10-s + 2.30·12-s − 2.21·13-s + 1.06·14-s − 2.06·15-s − 16-s + 0.970·17-s + 3.77·18-s − 0.917·19-s − 0.894·20-s + 1.74·21-s + 1.66·23-s − 1/5·25-s − 3.13·26-s + 2.30·27-s + 0.755·28-s − 2.92·30-s − 1.41·32-s + 1.37·34-s − 0.676·35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 48400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.719046670\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.719046670\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - p T + p T^{2} \) |
| 5 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 11 | $C_2$ | \( 1 + T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 8 T + 32 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 8 T + 32 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 54 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 8 T + 32 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 142 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
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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
−12.45985598901552531534436160613, −12.18867249337151837304572447647, −12.01700346542955690325834525975, −11.29397202068506082252926055128, −10.54137573775509692327993230978, −10.11648237488714637021480229585, −9.453633220649238396697070769593, −8.831921865954385221617692499110, −8.624312871136564018272125362135, −8.077176421466973270283863362218, −7.33597006180653478265023245178, −7.31245446689933418325187777012, −6.61635367314294036585971940962, −5.25738493624964375170500868039, −5.10946208062674339360294620042, −4.38338932882092220423139813221, −3.72139969892751743934681285446, −3.27808248789426262086213812530, −2.59057171889261465725008868322, −2.09092810389286353199831228164,
2.09092810389286353199831228164, 2.59057171889261465725008868322, 3.27808248789426262086213812530, 3.72139969892751743934681285446, 4.38338932882092220423139813221, 5.10946208062674339360294620042, 5.25738493624964375170500868039, 6.61635367314294036585971940962, 7.31245446689933418325187777012, 7.33597006180653478265023245178, 8.077176421466973270283863362218, 8.624312871136564018272125362135, 8.831921865954385221617692499110, 9.453633220649238396697070769593, 10.11648237488714637021480229585, 10.54137573775509692327993230978, 11.29397202068506082252926055128, 12.01700346542955690325834525975, 12.18867249337151837304572447647, 12.45985598901552531534436160613
