L(s) = 1 | − 2-s + 4-s − 7-s − 8-s + 1.73·11-s − 1.46·13-s + 14-s + 16-s − 4·17-s + 2.26·19-s − 1.73·22-s + 4.46·23-s + 1.46·26-s − 28-s + 5.46·29-s − 3.73·31-s − 32-s + 4·34-s + 7.19·37-s − 2.26·38-s − 9.92·41-s − 11.4·43-s + 1.73·44-s − 4.46·46-s − 4.53·47-s + 49-s − 1.46·52-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.5·4-s − 0.377·7-s − 0.353·8-s + 0.522·11-s − 0.406·13-s + 0.267·14-s + 0.250·16-s − 0.970·17-s + 0.520·19-s − 0.369·22-s + 0.930·23-s + 0.287·26-s − 0.188·28-s + 1.01·29-s − 0.670·31-s − 0.176·32-s + 0.685·34-s + 1.18·37-s − 0.367·38-s − 1.55·41-s − 1.74·43-s + 0.261·44-s − 0.658·46-s − 0.661·47-s + 0.142·49-s − 0.203·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
good | 11 | \( 1 - 1.73T + 11T^{2} \) |
| 13 | \( 1 + 1.46T + 13T^{2} \) |
| 17 | \( 1 + 4T + 17T^{2} \) |
| 19 | \( 1 - 2.26T + 19T^{2} \) |
| 23 | \( 1 - 4.46T + 23T^{2} \) |
| 29 | \( 1 - 5.46T + 29T^{2} \) |
| 31 | \( 1 + 3.73T + 31T^{2} \) |
| 37 | \( 1 - 7.19T + 37T^{2} \) |
| 41 | \( 1 + 9.92T + 41T^{2} \) |
| 43 | \( 1 + 11.4T + 43T^{2} \) |
| 47 | \( 1 + 4.53T + 47T^{2} \) |
| 53 | \( 1 + 13.8T + 53T^{2} \) |
| 59 | \( 1 - 8.92T + 59T^{2} \) |
| 61 | \( 1 - 13.8T + 61T^{2} \) |
| 67 | \( 1 - 8.39T + 67T^{2} \) |
| 71 | \( 1 + 7.73T + 71T^{2} \) |
| 73 | \( 1 - 3.07T + 73T^{2} \) |
| 79 | \( 1 + 4.92T + 79T^{2} \) |
| 83 | \( 1 + 2.53T + 83T^{2} \) |
| 89 | \( 1 + T + 89T^{2} \) |
| 97 | \( 1 - 6.92T + 97T^{2} \) |
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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
−7.26056615510754345583102193251, −6.75063540134483602635818183567, −6.32180130208424906633073065700, −5.26379787694193833579688367788, −4.68267731288575248036111201784, −3.64073268199423282284069418980, −2.96300025091358992346158668739, −2.07364431781472377069137795774, −1.13285182346092245010466261572, 0,
1.13285182346092245010466261572, 2.07364431781472377069137795774, 2.96300025091358992346158668739, 3.64073268199423282284069418980, 4.68267731288575248036111201784, 5.26379787694193833579688367788, 6.32180130208424906633073065700, 6.75063540134483602635818183567, 7.26056615510754345583102193251