L(s) = 1 | − 3·3-s − 4-s + 6·9-s − 3·11-s + 3·12-s + 16-s − 11·23-s − 9·27-s − 5·31-s + 9·33-s − 6·36-s + 3·37-s + 3·44-s + 3·47-s − 3·48-s − 49-s − 8·53-s + 6·59-s − 64-s − 24·67-s + 33·69-s − 18·71-s + 9·81-s + 3·89-s + 11·92-s + 15·93-s − 9·97-s + ⋯ |
L(s) = 1 | − 1.73·3-s − 1/2·4-s + 2·9-s − 0.904·11-s + 0.866·12-s + 1/4·16-s − 2.29·23-s − 1.73·27-s − 0.898·31-s + 1.56·33-s − 36-s + 0.493·37-s + 0.452·44-s + 0.437·47-s − 0.433·48-s − 1/7·49-s − 1.09·53-s + 0.781·59-s − 1/8·64-s − 2.93·67-s + 3.97·69-s − 2.13·71-s + 81-s + 0.317·89-s + 1.14·92-s + 1.55·93-s − 0.913·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2722500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2722500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + T^{2} \) |
| 3 | $C_2$ | \( 1 + p T + p T^{2} \) |
| 5 | | \( 1 \) |
| 11 | $C_2$ | \( 1 + 3 T + p T^{2} \) |
good | 7 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 15 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 32 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 + 12 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 72 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 + 65 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 + 10 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 32 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 156 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
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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
−7.07730273660053823793455827673, −6.74465279397683915960745273789, −6.09994197507349867252546842141, −5.83175420171485280975180151485, −5.66355886891953977288563234426, −5.12868928657834090377889063580, −4.61546816735627806929946485692, −4.29610347352703667401700766817, −3.92063886261047611009323989336, −3.20720362019016000160518925462, −2.53804347070958778540743181219, −1.79160155305405056100188452196, −1.23033755313549059437998861462, 0, 0,
1.23033755313549059437998861462, 1.79160155305405056100188452196, 2.53804347070958778540743181219, 3.20720362019016000160518925462, 3.92063886261047611009323989336, 4.29610347352703667401700766817, 4.61546816735627806929946485692, 5.12868928657834090377889063580, 5.66355886891953977288563234426, 5.83175420171485280975180151485, 6.09994197507349867252546842141, 6.74465279397683915960745273789, 7.07730273660053823793455827673