| L(s) = 1 | − 3-s + 2·7-s + 9-s + 2·11-s − 2·13-s − 6·17-s + 8·19-s − 2·21-s + 4·23-s − 27-s + 8·29-s − 2·33-s + 10·37-s + 2·39-s + 2·41-s + 12·43-s − 3·49-s + 6·51-s − 10·53-s − 8·57-s − 6·59-s + 2·61-s + 2·63-s + 8·67-s − 4·69-s − 4·71-s − 4·73-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.755·7-s + 1/3·9-s + 0.603·11-s − 0.554·13-s − 1.45·17-s + 1.83·19-s − 0.436·21-s + 0.834·23-s − 0.192·27-s + 1.48·29-s − 0.348·33-s + 1.64·37-s + 0.320·39-s + 0.312·41-s + 1.82·43-s − 3/7·49-s + 0.840·51-s − 1.37·53-s − 1.05·57-s − 0.781·59-s + 0.256·61-s + 0.251·63-s + 0.977·67-s − 0.481·69-s − 0.474·71-s − 0.468·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.364104752\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.364104752\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−10.93771228286082190203080606653, −9.742155904784536505330609626306, −9.059763928943775553795046458687, −7.904705684497857379917841291631, −7.07357706190252596248698932419, −6.13886816727698403174895074153, −5.00409208200375228507530236068, −4.34068578600963751988277087473, −2.73466259421800465396662803312, −1.14585721906037177235111339150,
1.14585721906037177235111339150, 2.73466259421800465396662803312, 4.34068578600963751988277087473, 5.00409208200375228507530236068, 6.13886816727698403174895074153, 7.07357706190252596248698932419, 7.904705684497857379917841291631, 9.059763928943775553795046458687, 9.742155904784536505330609626306, 10.93771228286082190203080606653
