L(s) = 1 | + 5-s − 2·7-s + 13-s − 2·19-s − 6·23-s + 25-s + 4·31-s − 2·35-s + 2·37-s + 6·41-s + 4·43-s − 3·49-s + 6·53-s − 10·61-s + 65-s − 8·67-s + 8·73-s − 8·79-s − 12·83-s − 6·89-s − 2·91-s − 2·95-s + 8·97-s + 12·101-s − 8·103-s + 12·107-s − 16·109-s + ⋯ |
L(s) = 1 | + 0.447·5-s − 0.755·7-s + 0.277·13-s − 0.458·19-s − 1.25·23-s + 1/5·25-s + 0.718·31-s − 0.338·35-s + 0.328·37-s + 0.937·41-s + 0.609·43-s − 3/7·49-s + 0.824·53-s − 1.28·61-s + 0.124·65-s − 0.977·67-s + 0.936·73-s − 0.900·79-s − 1.31·83-s − 0.635·89-s − 0.209·91-s − 0.205·95-s + 0.812·97-s + 1.19·101-s − 0.788·103-s + 1.16·107-s − 1.53·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 - T \) |
good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 8 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−7.39879715982559174388842602129, −6.43381734294611879067068272380, −6.16937951130498734311665955734, −5.45128516441390105434556892670, −4.47358233185303425172116418963, −3.88309069005629741914214277777, −2.95189431742758179873473786174, −2.27341836583634432376611855584, −1.23811175511269260606560380687, 0,
1.23811175511269260606560380687, 2.27341836583634432376611855584, 2.95189431742758179873473786174, 3.88309069005629741914214277777, 4.47358233185303425172116418963, 5.45128516441390105434556892670, 6.16937951130498734311665955734, 6.43381734294611879067068272380, 7.39879715982559174388842602129