| L(s) = 1 | − 2-s + 4-s − 4·5-s − 4·7-s − 8-s − 2·9-s + 4·10-s + 4·11-s + 10·13-s + 4·14-s + 16-s + 2·18-s − 4·20-s − 4·22-s + 11·25-s − 10·26-s − 4·28-s − 16·31-s − 32-s + 16·35-s − 2·36-s + 4·40-s + 4·44-s + 8·45-s + 9·49-s − 11·50-s + 10·52-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 1.78·5-s − 1.51·7-s − 0.353·8-s − 2/3·9-s + 1.26·10-s + 1.20·11-s + 2.77·13-s + 1.06·14-s + 1/4·16-s + 0.471·18-s − 0.894·20-s − 0.852·22-s + 11/5·25-s − 1.96·26-s − 0.755·28-s − 2.87·31-s − 0.176·32-s + 2.70·35-s − 1/3·36-s + 0.632·40-s + 0.603·44-s + 1.19·45-s + 9/7·49-s − 1.55·50-s + 1.38·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 156800 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 156800 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5783504312\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5783504312\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 + T \) |
| 5 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| 7 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| good | 3 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 - 42 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + 42 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 62 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
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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
−9.025694293612620508698369601370, −8.775869917029675650777173279152, −8.542899990789323607745426433184, −7.84567693863861201615669999618, −7.34632989273407511334351307479, −6.88020909301788921798556576454, −6.37804844909400536852667019536, −6.01179633321282772578481474537, −5.46643161204975463040257635687, −4.25705032924395433442249010201, −3.81419986888863923682638475518, −3.41882772768279606162894089044, −3.13112984015444259143376519045, −1.64424561543330803519132530159, −0.58736775770773719122505341873,
0.58736775770773719122505341873, 1.64424561543330803519132530159, 3.13112984015444259143376519045, 3.41882772768279606162894089044, 3.81419986888863923682638475518, 4.25705032924395433442249010201, 5.46643161204975463040257635687, 6.01179633321282772578481474537, 6.37804844909400536852667019536, 6.88020909301788921798556576454, 7.34632989273407511334351307479, 7.84567693863861201615669999618, 8.542899990789323607745426433184, 8.775869917029675650777173279152, 9.025694293612620508698369601370
