L(s) = 1 | − 2·3-s − 7-s + 9-s − 3·11-s + 4·13-s + 3·17-s − 19-s + 2·21-s + 4·27-s + 6·29-s + 4·31-s + 6·33-s − 2·37-s − 8·39-s − 6·41-s − 43-s − 3·47-s − 6·49-s − 6·51-s − 12·53-s + 2·57-s + 6·59-s − 61-s − 63-s − 4·67-s − 6·71-s + 7·73-s + ⋯ |
L(s) = 1 | − 1.15·3-s − 0.377·7-s + 1/3·9-s − 0.904·11-s + 1.10·13-s + 0.727·17-s − 0.229·19-s + 0.436·21-s + 0.769·27-s + 1.11·29-s + 0.718·31-s + 1.04·33-s − 0.328·37-s − 1.28·39-s − 0.937·41-s − 0.152·43-s − 0.437·47-s − 6/7·49-s − 0.840·51-s − 1.64·53-s + 0.264·57-s + 0.781·59-s − 0.128·61-s − 0.125·63-s − 0.488·67-s − 0.712·71-s + 0.819·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9228459137\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9228459137\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 + T \) |
good | 3 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - 7 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 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
−8.051835152099631546993687645092, −6.90506256205873325459925429282, −6.39631576546855752821421102492, −5.84571883696030556656372119783, −5.13832245086405846361036626372, −4.57629137159943570016406304260, −3.46517685806800027733679125643, −2.84164120918815306754360859278, −1.53581733995414119392489635744, −0.52799044583343788332623178895,
0.52799044583343788332623178895, 1.53581733995414119392489635744, 2.84164120918815306754360859278, 3.46517685806800027733679125643, 4.57629137159943570016406304260, 5.13832245086405846361036626372, 5.84571883696030556656372119783, 6.39631576546855752821421102492, 6.90506256205873325459925429282, 8.051835152099631546993687645092