L(s) = 1 | − 2.96·2-s + 0.788·4-s + 20.3·5-s − 25.7·7-s + 21.3·8-s − 60.1·10-s − 49.6·11-s + 51.2·13-s + 76.1·14-s − 69.6·16-s − 127.·17-s − 97.6·19-s + 16.0·20-s + 147.·22-s − 175.·23-s + 287.·25-s − 151.·26-s − 20.2·28-s − 16.4·29-s − 289.·31-s + 35.5·32-s + 378.·34-s − 521.·35-s + 86.9·37-s + 289.·38-s + 434.·40-s − 88.5·41-s + ⋯ |
L(s) = 1 | − 1.04·2-s + 0.0985·4-s + 1.81·5-s − 1.38·7-s + 0.944·8-s − 1.90·10-s − 1.35·11-s + 1.09·13-s + 1.45·14-s − 1.08·16-s − 1.82·17-s − 1.17·19-s + 0.178·20-s + 1.42·22-s − 1.59·23-s + 2.29·25-s − 1.14·26-s − 0.136·28-s − 0.105·29-s − 1.67·31-s + 0.196·32-s + 1.90·34-s − 2.52·35-s + 0.386·37-s + 1.23·38-s + 1.71·40-s − 0.337·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.5652218001\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5652218001\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 239 | \( 1 + 239T \) |
good | 2 | \( 1 + 2.96T + 8T^{2} \) |
| 5 | \( 1 - 20.3T + 125T^{2} \) |
| 7 | \( 1 + 25.7T + 343T^{2} \) |
| 11 | \( 1 + 49.6T + 1.33e3T^{2} \) |
| 13 | \( 1 - 51.2T + 2.19e3T^{2} \) |
| 17 | \( 1 + 127.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 97.6T + 6.85e3T^{2} \) |
| 23 | \( 1 + 175.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 16.4T + 2.43e4T^{2} \) |
| 31 | \( 1 + 289.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 86.9T + 5.06e4T^{2} \) |
| 41 | \( 1 + 88.5T + 6.89e4T^{2} \) |
| 43 | \( 1 + 346.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 346.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 234.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 427.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 170.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 724.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 29.5T + 3.57e5T^{2} \) |
| 73 | \( 1 + 1.20e3T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.20e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 191.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.00e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 58.7T + 9.12e5T^{2} \) |
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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
−8.877705086076100655638849894706, −8.335155383150535706052600555592, −7.12162354487654802856055642652, −6.33795019665408062234708113308, −5.89787943329636133250509751682, −4.85457004542000663516640623405, −3.69839625891451147330547092690, −2.29449261611501478818301680869, −1.93619076213978370805313021849, −0.37862981426351275850846076241,
0.37862981426351275850846076241, 1.93619076213978370805313021849, 2.29449261611501478818301680869, 3.69839625891451147330547092690, 4.85457004542000663516640623405, 5.89787943329636133250509751682, 6.33795019665408062234708113308, 7.12162354487654802856055642652, 8.335155383150535706052600555592, 8.877705086076100655638849894706