| L(s) = 1 | − 2·7-s + 11-s + 13-s + 2·17-s + 6·19-s − 8·23-s − 5·25-s + 2·29-s − 6·31-s − 6·37-s + 8·41-s − 8·43-s − 12·47-s − 3·49-s + 10·53-s + 8·59-s + 2·61-s + 6·67-s + 8·71-s − 2·73-s − 2·77-s + 12·79-s − 8·83-s − 2·91-s − 18·97-s + 18·101-s − 16·103-s + ⋯ |
| L(s) = 1 | − 0.755·7-s + 0.301·11-s + 0.277·13-s + 0.485·17-s + 1.37·19-s − 1.66·23-s − 25-s + 0.371·29-s − 1.07·31-s − 0.986·37-s + 1.24·41-s − 1.21·43-s − 1.75·47-s − 3/7·49-s + 1.37·53-s + 1.04·59-s + 0.256·61-s + 0.733·67-s + 0.949·71-s − 0.234·73-s − 0.227·77-s + 1.35·79-s − 0.878·83-s − 0.209·91-s − 1.82·97-s + 1.79·101-s − 1.57·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5148 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 6 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
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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
−7.891696439983347553150977979253, −7.11823433160350637181889500236, −6.41233655133032191234737902789, −5.70784284827979668740879792565, −5.07407820840454882496452191222, −3.77332098383130374353748205226, −3.56934235983159125897681134704, −2.38568013392244188587309177745, −1.35551607999366076877403044600, 0,
1.35551607999366076877403044600, 2.38568013392244188587309177745, 3.56934235983159125897681134704, 3.77332098383130374353748205226, 5.07407820840454882496452191222, 5.70784284827979668740879792565, 6.41233655133032191234737902789, 7.11823433160350637181889500236, 7.891696439983347553150977979253
