| L(s) = 1 | + 3-s − 4·7-s + 9-s + 11-s − 13-s + 8·17-s − 4·19-s − 4·21-s − 5·25-s + 27-s + 4·29-s − 2·31-s + 33-s − 2·37-s − 39-s − 2·41-s + 2·43-s − 4·47-s + 9·49-s + 8·51-s + 6·53-s − 4·57-s − 10·61-s − 4·63-s − 2·67-s − 14·73-s − 5·75-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.51·7-s + 1/3·9-s + 0.301·11-s − 0.277·13-s + 1.94·17-s − 0.917·19-s − 0.872·21-s − 25-s + 0.192·27-s + 0.742·29-s − 0.359·31-s + 0.174·33-s − 0.328·37-s − 0.160·39-s − 0.312·41-s + 0.304·43-s − 0.583·47-s + 9/7·49-s + 1.12·51-s + 0.824·53-s − 0.529·57-s − 1.28·61-s − 0.503·63-s − 0.244·67-s − 1.63·73-s − 0.577·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 8 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 16 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.56782286225328193112264934939, −7.00219669551555731832290169226, −6.17239055563508803098187886902, −5.71991945113624794291730067509, −4.62352960148066102606067520868, −3.71373317511754436142355942160, −3.25997672277555714769663608301, −2.46203922830889927495051900442, −1.32721688032948586871388779415, 0,
1.32721688032948586871388779415, 2.46203922830889927495051900442, 3.25997672277555714769663608301, 3.71373317511754436142355942160, 4.62352960148066102606067520868, 5.71991945113624794291730067509, 6.17239055563508803098187886902, 7.00219669551555731832290169226, 7.56782286225328193112264934939
