L(s) = 1 | + 3-s − 5-s − 7-s + 9-s − 2·13-s − 15-s − 6·17-s + 8·19-s − 21-s + 25-s + 27-s − 6·29-s + 4·31-s + 35-s + 10·37-s − 2·39-s − 6·41-s − 4·43-s − 45-s + 49-s − 6·51-s + 6·53-s + 8·57-s − 12·59-s + 10·61-s − 63-s + 2·65-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 0.447·5-s − 0.377·7-s + 1/3·9-s − 0.554·13-s − 0.258·15-s − 1.45·17-s + 1.83·19-s − 0.218·21-s + 1/5·25-s + 0.192·27-s − 1.11·29-s + 0.718·31-s + 0.169·35-s + 1.64·37-s − 0.320·39-s − 0.937·41-s − 0.609·43-s − 0.149·45-s + 1/7·49-s − 0.840·51-s + 0.824·53-s + 1.05·57-s − 1.56·59-s + 1.28·61-s − 0.125·63-s + 0.248·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 + T \) |
good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 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 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−7.56082479581924086601461922273, −7.12859150809063645513499611375, −6.35459912228135431115432186201, −5.44674315289696763385051199981, −4.63966351865398706236897283548, −3.96385392514530739831823934957, −3.09459576654432180909497533638, −2.49000324770899681368855710467, −1.33618579822246687901821912080, 0,
1.33618579822246687901821912080, 2.49000324770899681368855710467, 3.09459576654432180909497533638, 3.96385392514530739831823934957, 4.63966351865398706236897283548, 5.44674315289696763385051199981, 6.35459912228135431115432186201, 7.12859150809063645513499611375, 7.56082479581924086601461922273