| L(s) = 1 | − 3·3-s − 3·4-s − 5-s + 6·9-s + 3·11-s + 9·12-s + 3·15-s + 5·16-s + 3·20-s + 7·23-s − 7·25-s − 9·27-s − 5·31-s − 9·33-s − 18·36-s − 3·37-s − 9·44-s − 6·45-s + 5·47-s − 15·48-s − 11·49-s − 5·53-s − 3·55-s − 3·59-s − 9·60-s − 3·64-s − 6·67-s + ⋯ |
| L(s) = 1 | − 1.73·3-s − 3/2·4-s − 0.447·5-s + 2·9-s + 0.904·11-s + 2.59·12-s + 0.774·15-s + 5/4·16-s + 0.670·20-s + 1.45·23-s − 7/5·25-s − 1.73·27-s − 0.898·31-s − 1.56·33-s − 3·36-s − 0.493·37-s − 1.35·44-s − 0.894·45-s + 0.729·47-s − 2.16·48-s − 1.57·49-s − 0.686·53-s − 0.404·55-s − 0.390·59-s − 1.16·60-s − 3/8·64-s − 0.733·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25047 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25047 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | $C_2$ | \( 1 + p T + p T^{2} \) |
| 11 | $C_2$ | \( 1 - 3 T + p T^{2} \) |
| 23 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 - 6 T + p T^{2} ) \) |
| good | 2 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 5 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2^2$ | \( 1 + 11 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 15 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 31 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 55 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 41 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 + 114 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 + 12 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 - 13 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 143 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 41 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
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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
−10.45715402421865550017158147261, −9.907813772532039625194893482897, −9.354255600192595666072923996922, −8.987334907847032372266694963918, −8.355452370493371975346300152634, −7.47545237040884299028314839664, −7.16275471097725371864787753723, −6.22985218533883797555173099994, −5.89850219230318835555125348726, −5.06005851722607127975323573237, −4.73102153871470454016218385864, −4.08241272011850993008377291555, −3.44170186527149644991256333227, −1.38142750019945230118042835170, 0,
1.38142750019945230118042835170, 3.44170186527149644991256333227, 4.08241272011850993008377291555, 4.73102153871470454016218385864, 5.06005851722607127975323573237, 5.89850219230318835555125348726, 6.22985218533883797555173099994, 7.16275471097725371864787753723, 7.47545237040884299028314839664, 8.355452370493371975346300152634, 8.987334907847032372266694963918, 9.354255600192595666072923996922, 9.907813772532039625194893482897, 10.45715402421865550017158147261
