L(s) = 1 | + 3-s − 5-s − 2·7-s + 9-s − 2·13-s − 15-s − 2·19-s − 2·21-s + 25-s + 27-s + 8·31-s + 2·35-s + 2·37-s − 2·39-s − 2·43-s − 45-s − 3·49-s + 6·53-s − 2·57-s − 12·59-s − 2·61-s − 2·63-s + 2·65-s − 4·67-s − 2·73-s + 75-s + 10·79-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 0.447·5-s − 0.755·7-s + 1/3·9-s − 0.554·13-s − 0.258·15-s − 0.458·19-s − 0.436·21-s + 1/5·25-s + 0.192·27-s + 1.43·31-s + 0.338·35-s + 0.328·37-s − 0.320·39-s − 0.304·43-s − 0.149·45-s − 3/7·49-s + 0.824·53-s − 0.264·57-s − 1.56·59-s − 0.256·61-s − 0.251·63-s + 0.248·65-s − 0.488·67-s − 0.234·73-s + 0.115·75-s + 1.12·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.709283426\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.709283426\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 \) |
good | 7 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−7.923407605983994250589950629717, −7.30006072679712949741046823488, −6.57103895258206465198836329947, −5.99832161434329141032732583940, −4.87760095706776241965692070574, −4.33147484644671132179144222503, −3.41292940185586984061180216922, −2.85372106253460268189300561756, −1.94220370965866716469868347087, −0.62346580820178742493032848749,
0.62346580820178742493032848749, 1.94220370965866716469868347087, 2.85372106253460268189300561756, 3.41292940185586984061180216922, 4.33147484644671132179144222503, 4.87760095706776241965692070574, 5.99832161434329141032732583940, 6.57103895258206465198836329947, 7.30006072679712949741046823488, 7.923407605983994250589950629717