| L(s) = 1 | − 2·3-s − 4-s + 3·9-s + 2·12-s − 3·16-s + 25-s − 4·27-s + 8·31-s − 3·36-s + 8·37-s + 12·47-s + 6·48-s − 2·49-s − 12·59-s + 7·64-s + 8·67-s − 12·71-s − 2·75-s + 5·81-s − 16·93-s + 4·97-s − 100-s + 4·103-s + 4·108-s − 16·111-s − 12·113-s − 8·124-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 1/2·4-s + 9-s + 0.577·12-s − 3/4·16-s + 1/5·25-s − 0.769·27-s + 1.43·31-s − 1/2·36-s + 1.31·37-s + 1.75·47-s + 0.866·48-s − 2/7·49-s − 1.56·59-s + 7/8·64-s + 0.977·67-s − 1.42·71-s − 0.230·75-s + 5/9·81-s − 1.65·93-s + 0.406·97-s − 0.0999·100-s + 0.394·103-s + 0.384·108-s − 1.51·111-s − 1.12·113-s − 0.718·124-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_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 11 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 - 94 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
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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
−7.26865453813587007648236585531, −6.85148596046844826148453491842, −6.30895003848223476906417348596, −6.10393323956121282521751877576, −5.73064095886285491692068937765, −5.09249574448959305810139200699, −4.77801578706496934630419848355, −4.44994679873204421798537317144, −4.02803965188681937843711153412, −3.48353662209482708119770478632, −2.68072355101008303033232110220, −2.35824350904084673016574405264, −1.40326365182592763485746000445, −0.848684206917762978894320143348, 0,
0.848684206917762978894320143348, 1.40326365182592763485746000445, 2.35824350904084673016574405264, 2.68072355101008303033232110220, 3.48353662209482708119770478632, 4.02803965188681937843711153412, 4.44994679873204421798537317144, 4.77801578706496934630419848355, 5.09249574448959305810139200699, 5.73064095886285491692068937765, 6.10393323956121282521751877576, 6.30895003848223476906417348596, 6.85148596046844826148453491842, 7.26865453813587007648236585531
