| L(s) = 1 | − 2·3-s + 4-s + 2·5-s + 3·9-s − 2·12-s − 4·15-s − 3·16-s + 2·20-s + 10·23-s − 25-s − 4·27-s − 2·31-s + 3·36-s − 6·37-s + 6·45-s − 2·47-s + 6·48-s + 8·49-s + 10·53-s − 18·59-s − 4·60-s − 7·64-s − 4·67-s − 20·69-s + 2·75-s − 6·80-s + 5·81-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 1/2·4-s + 0.894·5-s + 9-s − 0.577·12-s − 1.03·15-s − 3/4·16-s + 0.447·20-s + 2.08·23-s − 1/5·25-s − 0.769·27-s − 0.359·31-s + 1/2·36-s − 0.986·37-s + 0.894·45-s − 0.291·47-s + 0.866·48-s + 8/7·49-s + 1.37·53-s − 2.34·59-s − 0.516·60-s − 7/8·64-s − 0.488·67-s − 2.40·69-s + 0.230·75-s − 0.670·80-s + 5/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3294225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3294225 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 5 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| 11 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 54 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 4 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 + 80 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
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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
−7.23473619210511184343399233246, −6.76232375623165558787477433357, −6.47329759783796833964144798890, −6.14351387784493089419951352939, −5.50171559504015289102333086195, −5.38207188464990178748829781459, −4.94008125660946637735343919034, −4.41086721376393163602033352362, −3.97436273141080246034087263155, −3.23878614191063938806341448105, −2.75749348557958634561676664328, −2.16920294421874178207978396532, −1.59081461334933050296525140582, −1.04172320822173386745969461544, 0,
1.04172320822173386745969461544, 1.59081461334933050296525140582, 2.16920294421874178207978396532, 2.75749348557958634561676664328, 3.23878614191063938806341448105, 3.97436273141080246034087263155, 4.41086721376393163602033352362, 4.94008125660946637735343919034, 5.38207188464990178748829781459, 5.50171559504015289102333086195, 6.14351387784493089419951352939, 6.47329759783796833964144798890, 6.76232375623165558787477433357, 7.23473619210511184343399233246
