| L(s) = 1 | + 3-s + 5·7-s + 9-s − 2·11-s − 13-s − 4·17-s − 7·19-s + 5·21-s − 6·23-s + 27-s − 6·29-s − 31-s − 2·33-s − 37-s − 39-s − 2·41-s + 11·43-s + 4·47-s + 18·49-s − 4·51-s + 4·53-s − 7·57-s + 10·59-s + 5·61-s + 5·63-s + 5·67-s − 6·69-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.88·7-s + 1/3·9-s − 0.603·11-s − 0.277·13-s − 0.970·17-s − 1.60·19-s + 1.09·21-s − 1.25·23-s + 0.192·27-s − 1.11·29-s − 0.179·31-s − 0.348·33-s − 0.164·37-s − 0.160·39-s − 0.312·41-s + 1.67·43-s + 0.583·47-s + 18/7·49-s − 0.560·51-s + 0.549·53-s − 0.927·57-s + 1.30·59-s + 0.640·61-s + 0.629·63-s + 0.610·67-s − 0.722·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 44400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 44400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.913553061\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.913553061\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 37 | \( 1 + T \) |
| good | 7 | \( 1 - 5 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 7 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 11 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 97 | \( 1 + T + p T^{2} \) |
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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
−14.60691262051354, −14.36743836680403, −13.59545830113960, −13.29654222657263, −12.56712259849277, −12.15754834665301, −11.37869762077467, −11.05319691675027, −10.56324806263770, −10.06498448935195, −9.230709141499600, −8.721591400411172, −8.304431944237911, −7.854286026970021, −7.361119247977247, −6.735195075195224, −5.893338103320280, −5.351836220340223, −4.721214258931224, −4.118655805679376, −3.836890659911953, −2.430622908622041, −2.247125633054435, −1.695402980797156, −0.5631371510326130,
0.5631371510326130, 1.695402980797156, 2.247125633054435, 2.430622908622041, 3.836890659911953, 4.118655805679376, 4.721214258931224, 5.351836220340223, 5.893338103320280, 6.735195075195224, 7.361119247977247, 7.854286026970021, 8.304431944237911, 8.721591400411172, 9.230709141499600, 10.06498448935195, 10.56324806263770, 11.05319691675027, 11.37869762077467, 12.15754834665301, 12.56712259849277, 13.29654222657263, 13.59545830113960, 14.36743836680403, 14.60691262051354
