L(s) = 1 | − 2·13-s + 8·19-s − 5·25-s − 4·31-s − 10·37-s − 8·43-s − 14·61-s + 16·67-s + 10·73-s + 4·79-s − 14·97-s + 20·103-s + 2·109-s + ⋯ |
L(s) = 1 | − 0.554·13-s + 1.83·19-s − 25-s − 0.718·31-s − 1.64·37-s − 1.21·43-s − 1.79·61-s + 1.95·67-s + 1.17·73-s + 0.450·79-s − 1.42·97-s + 1.97·103-s + 0.191·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 - 16 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−7.55348034334582611202573667047, −7.01678164134878036064700533521, −6.19926344367899340966030638418, −5.29098134252625465872380406245, −4.99119743714055460794390382794, −3.79768693875931379182391471043, −3.27428940642722031982966070935, −2.24166501763825187887414431892, −1.32729943198718475773059677845, 0,
1.32729943198718475773059677845, 2.24166501763825187887414431892, 3.27428940642722031982966070935, 3.79768693875931379182391471043, 4.99119743714055460794390382794, 5.29098134252625465872380406245, 6.19926344367899340966030638418, 7.01678164134878036064700533521, 7.55348034334582611202573667047