L(s) = 1 | + 2-s + 3.18·3-s + 4-s − 5-s + 3.18·6-s − 7-s + 8-s + 7.16·9-s − 10-s + 3.18·12-s + 5.69·13-s − 14-s − 3.18·15-s + 16-s − 0.749·17-s + 7.16·18-s + 6.78·19-s − 20-s − 3.18·21-s + 3.94·23-s + 3.18·24-s + 25-s + 5.69·26-s + 13.2·27-s − 28-s − 5.73·29-s − 3.18·30-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1.84·3-s + 0.5·4-s − 0.447·5-s + 1.30·6-s − 0.377·7-s + 0.353·8-s + 2.38·9-s − 0.316·10-s + 0.920·12-s + 1.57·13-s − 0.267·14-s − 0.823·15-s + 0.250·16-s − 0.181·17-s + 1.68·18-s + 1.55·19-s − 0.223·20-s − 0.695·21-s + 0.821·23-s + 0.650·24-s + 0.200·25-s + 1.11·26-s + 2.55·27-s − 0.188·28-s − 1.06·29-s − 0.581·30-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
\(L(1)\) |
\(\approx\) |
\(7.072441461\) |
\(L(\frac12)\) |
\(\approx\) |
\(7.072441461\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - 3.18T + 3T^{2} \) |
| 13 | \( 1 - 5.69T + 13T^{2} \) |
| 17 | \( 1 + 0.749T + 17T^{2} \) |
| 19 | \( 1 - 6.78T + 19T^{2} \) |
| 23 | \( 1 - 3.94T + 23T^{2} \) |
| 29 | \( 1 + 5.73T + 29T^{2} \) |
| 31 | \( 1 + 7.10T + 31T^{2} \) |
| 37 | \( 1 + 11.6T + 37T^{2} \) |
| 41 | \( 1 - 11.4T + 41T^{2} \) |
| 43 | \( 1 - 2.33T + 43T^{2} \) |
| 47 | \( 1 + 5.00T + 47T^{2} \) |
| 53 | \( 1 + 10.7T + 53T^{2} \) |
| 59 | \( 1 - 5.22T + 59T^{2} \) |
| 61 | \( 1 - 7.68T + 61T^{2} \) |
| 67 | \( 1 + 7.39T + 67T^{2} \) |
| 71 | \( 1 + 0.575T + 71T^{2} \) |
| 73 | \( 1 + 12.5T + 73T^{2} \) |
| 79 | \( 1 - 6.83T + 79T^{2} \) |
| 83 | \( 1 - 10.5T + 83T^{2} \) |
| 89 | \( 1 - 4.80T + 89T^{2} \) |
| 97 | \( 1 - 16.2T + 97T^{2} \) |
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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
−7.55965869144701947409529311549, −7.41220720340725167561629154586, −6.55369310984045896775163737649, −5.65539605283349356356340380832, −4.79486638996496190547738463558, −3.79434862675326071176715976063, −3.53065380777963496499476912937, −3.01394940324880800209783699245, −1.98494038726488819603554362848, −1.17615636695872939995265826848,
1.17615636695872939995265826848, 1.98494038726488819603554362848, 3.01394940324880800209783699245, 3.53065380777963496499476912937, 3.79434862675326071176715976063, 4.79486638996496190547738463558, 5.65539605283349356356340380832, 6.55369310984045896775163737649, 7.41220720340725167561629154586, 7.55965869144701947409529311549