| L(s) = 1 | + 3-s + 2·5-s − 2·7-s + 9-s + 11-s + 6·13-s + 2·15-s − 4·17-s + 2·19-s − 2·21-s + 8·23-s − 25-s + 27-s + 33-s − 4·35-s − 6·37-s + 6·39-s − 10·43-s + 2·45-s − 3·49-s − 4·51-s + 14·53-s + 2·55-s + 2·57-s + 12·59-s − 14·61-s − 2·63-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.894·5-s − 0.755·7-s + 1/3·9-s + 0.301·11-s + 1.66·13-s + 0.516·15-s − 0.970·17-s + 0.458·19-s − 0.436·21-s + 1.66·23-s − 1/5·25-s + 0.192·27-s + 0.174·33-s − 0.676·35-s − 0.986·37-s + 0.960·39-s − 1.52·43-s + 0.298·45-s − 3/7·49-s − 0.560·51-s + 1.92·53-s + 0.269·55-s + 0.264·57-s + 1.56·59-s − 1.79·61-s − 0.251·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 528 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.990658435\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.990658435\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 14 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 2 T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 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
−10.71879143206400174855148639397, −9.853168983015714730321615201563, −9.019094517389160161859110526078, −8.520481971363137668690428495851, −7.02339560430258550540873229128, −6.38383875473402815541456430546, −5.34474711434827294650147398392, −3.90041657716003839041565024444, −2.91854564551489641468247090209, −1.49985313315727000952810842633,
1.49985313315727000952810842633, 2.91854564551489641468247090209, 3.90041657716003839041565024444, 5.34474711434827294650147398392, 6.38383875473402815541456430546, 7.02339560430258550540873229128, 8.520481971363137668690428495851, 9.019094517389160161859110526078, 9.853168983015714730321615201563, 10.71879143206400174855148639397
