L(s) = 1 | − 3-s + 9-s − 2·11-s − 4·13-s − 27-s + 2·29-s + 10·31-s + 2·33-s − 4·37-s + 4·39-s + 6·41-s + 4·43-s − 7·49-s − 2·53-s − 6·59-s + 6·61-s + 4·67-s + 8·71-s − 14·73-s + 2·79-s + 81-s + 4·83-s − 2·87-s + 14·89-s − 10·93-s + 6·97-s − 2·99-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 1/3·9-s − 0.603·11-s − 1.10·13-s − 0.192·27-s + 0.371·29-s + 1.79·31-s + 0.348·33-s − 0.657·37-s + 0.640·39-s + 0.937·41-s + 0.609·43-s − 49-s − 0.274·53-s − 0.781·59-s + 0.768·61-s + 0.488·67-s + 0.949·71-s − 1.63·73-s + 0.225·79-s + 1/9·81-s + 0.439·83-s − 0.214·87-s + 1.48·89-s − 1.03·93-s + 0.609·97-s − 0.201·99-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - 2 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−7.42137206045731387956571558615, −6.55143673846532172969865055424, −6.08196247337318922483529380318, −5.08202663433445960270772560552, −4.83478249552697908292050480823, −3.93632783998710463972534562493, −2.89500726761865448274516487337, −2.26435217617127305427134080862, −1.07146533574569719267393540090, 0,
1.07146533574569719267393540090, 2.26435217617127305427134080862, 2.89500726761865448274516487337, 3.93632783998710463972534562493, 4.83478249552697908292050480823, 5.08202663433445960270772560552, 6.08196247337318922483529380318, 6.55143673846532172969865055424, 7.42137206045731387956571558615