L(s) = 1 | + 2-s + 2.86·3-s + 4-s − 5-s + 2.86·6-s + 4.06·7-s + 8-s + 5.19·9-s − 10-s + 11-s + 2.86·12-s + 7.12·13-s + 4.06·14-s − 2.86·15-s + 16-s − 7.00·17-s + 5.19·18-s − 2.52·19-s − 20-s + 11.6·21-s + 22-s + 0.631·23-s + 2.86·24-s + 25-s + 7.12·26-s + 6.27·27-s + 4.06·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1.65·3-s + 0.5·4-s − 0.447·5-s + 1.16·6-s + 1.53·7-s + 0.353·8-s + 1.73·9-s − 0.316·10-s + 0.301·11-s + 0.826·12-s + 1.97·13-s + 1.08·14-s − 0.739·15-s + 0.250·16-s − 1.69·17-s + 1.22·18-s − 0.580·19-s − 0.223·20-s + 2.53·21-s + 0.213·22-s + 0.131·23-s + 0.584·24-s + 0.200·25-s + 1.39·26-s + 1.20·27-s + 0.768·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4730 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4730 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(6.766639031\) |
\(L(\frac12)\) |
\(\approx\) |
\(6.766639031\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 43 | \( 1 - T \) |
good | 3 | \( 1 - 2.86T + 3T^{2} \) |
| 7 | \( 1 - 4.06T + 7T^{2} \) |
| 13 | \( 1 - 7.12T + 13T^{2} \) |
| 17 | \( 1 + 7.00T + 17T^{2} \) |
| 19 | \( 1 + 2.52T + 19T^{2} \) |
| 23 | \( 1 - 0.631T + 23T^{2} \) |
| 29 | \( 1 + 7.48T + 29T^{2} \) |
| 31 | \( 1 - 7.79T + 31T^{2} \) |
| 37 | \( 1 + 10.1T + 37T^{2} \) |
| 41 | \( 1 - 7.20T + 41T^{2} \) |
| 47 | \( 1 + 9.32T + 47T^{2} \) |
| 53 | \( 1 + 4.94T + 53T^{2} \) |
| 59 | \( 1 - 1.57T + 59T^{2} \) |
| 61 | \( 1 + 6.99T + 61T^{2} \) |
| 67 | \( 1 + 3.51T + 67T^{2} \) |
| 71 | \( 1 - 4.73T + 71T^{2} \) |
| 73 | \( 1 - 1.76T + 73T^{2} \) |
| 79 | \( 1 - 8.00T + 79T^{2} \) |
| 83 | \( 1 + 4.96T + 83T^{2} \) |
| 89 | \( 1 - 0.890T + 89T^{2} \) |
| 97 | \( 1 - 5.89T + 97T^{2} \) |
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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
−8.281152168475093471020083447696, −7.83341849327652624731572072284, −6.90694912936124427925215338581, −6.23678423951110435532912899552, −5.05830647581938641370760654905, −4.20387860193371834726128559970, −3.93157914366835380234073510550, −2.99892425702818867072882330067, −1.99234549502903730066889180998, −1.45973367727438874898308463533,
1.45973367727438874898308463533, 1.99234549502903730066889180998, 2.99892425702818867072882330067, 3.93157914366835380234073510550, 4.20387860193371834726128559970, 5.05830647581938641370760654905, 6.23678423951110435532912899552, 6.90694912936124427925215338581, 7.83341849327652624731572072284, 8.281152168475093471020083447696