L(s) = 1 | + 2-s + 0.529·3-s + 4-s + 5-s + 0.529·6-s + 7-s + 8-s − 2.71·9-s + 10-s + 0.529·12-s − 4.81·13-s + 14-s + 0.529·15-s + 16-s + 6.02·17-s − 2.71·18-s + 3.59·19-s + 20-s + 0.529·21-s − 7.06·23-s + 0.529·24-s + 25-s − 4.81·26-s − 3.02·27-s + 28-s + 0.353·29-s + 0.529·30-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.305·3-s + 0.5·4-s + 0.447·5-s + 0.216·6-s + 0.377·7-s + 0.353·8-s − 0.906·9-s + 0.316·10-s + 0.152·12-s − 1.33·13-s + 0.267·14-s + 0.136·15-s + 0.250·16-s + 1.46·17-s − 0.641·18-s + 0.824·19-s + 0.223·20-s + 0.115·21-s − 1.47·23-s + 0.108·24-s + 0.200·25-s − 0.943·26-s − 0.582·27-s + 0.188·28-s + 0.0656·29-s + 0.0966·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.943996001\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.943996001\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
good | 3 | \( 1 - 0.529T + 3T^{2} \) |
| 13 | \( 1 + 4.81T + 13T^{2} \) |
| 17 | \( 1 - 6.02T + 17T^{2} \) |
| 19 | \( 1 - 3.59T + 19T^{2} \) |
| 23 | \( 1 + 7.06T + 23T^{2} \) |
| 29 | \( 1 - 0.353T + 29T^{2} \) |
| 31 | \( 1 + 2.24T + 31T^{2} \) |
| 37 | \( 1 - 1.64T + 37T^{2} \) |
| 41 | \( 1 - 9.35T + 41T^{2} \) |
| 43 | \( 1 - 9.13T + 43T^{2} \) |
| 47 | \( 1 - 10.7T + 47T^{2} \) |
| 53 | \( 1 - 4.64T + 53T^{2} \) |
| 59 | \( 1 - 2.78T + 59T^{2} \) |
| 61 | \( 1 + 14.6T + 61T^{2} \) |
| 67 | \( 1 - 9.43T + 67T^{2} \) |
| 71 | \( 1 - 12.2T + 71T^{2} \) |
| 73 | \( 1 - 5.95T + 73T^{2} \) |
| 79 | \( 1 - 8.04T + 79T^{2} \) |
| 83 | \( 1 + 16.6T + 83T^{2} \) |
| 89 | \( 1 - 16.0T + 89T^{2} \) |
| 97 | \( 1 + 1.23T + 97T^{2} \) |
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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
−7.65529851659845534426940375915, −7.28180051700540984561661143797, −6.08919845880805791789458421155, −5.66448522170890462662845968475, −5.15372238653911817529811107517, −4.24568941968420146851520158895, −3.47136102177913093919227738784, −2.60529867143654073414288384609, −2.14410774574026698816559044689, −0.851192115674593447286541213651,
0.851192115674593447286541213651, 2.14410774574026698816559044689, 2.60529867143654073414288384609, 3.47136102177913093919227738784, 4.24568941968420146851520158895, 5.15372238653911817529811107517, 5.66448522170890462662845968475, 6.08919845880805791789458421155, 7.28180051700540984561661143797, 7.65529851659845534426940375915