| L(s) = 1 | − 2.67·2-s − 2.19·3-s + 5.13·4-s + 1.01·5-s + 5.85·6-s − 2.59·7-s − 8.35·8-s + 1.80·9-s − 2.69·10-s − 11-s − 11.2·12-s − 5.69·13-s + 6.93·14-s − 2.21·15-s + 12.0·16-s − 6.20·17-s − 4.82·18-s − 1.88·19-s + 5.18·20-s + 5.69·21-s + 2.67·22-s + 0.355·23-s + 18.3·24-s − 3.97·25-s + 15.1·26-s + 2.61·27-s − 13.3·28-s + ⋯ |
| L(s) = 1 | − 1.88·2-s − 1.26·3-s + 2.56·4-s + 0.452·5-s + 2.38·6-s − 0.981·7-s − 2.95·8-s + 0.601·9-s − 0.853·10-s − 0.301·11-s − 3.24·12-s − 1.57·13-s + 1.85·14-s − 0.572·15-s + 3.01·16-s − 1.50·17-s − 1.13·18-s − 0.431·19-s + 1.15·20-s + 1.24·21-s + 0.569·22-s + 0.0740·23-s + 3.74·24-s − 0.795·25-s + 2.97·26-s + 0.503·27-s − 2.51·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1005683987\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1005683987\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + T \) |
| 131 | \( 1 - T \) |
| good | 2 | \( 1 + 2.67T + 2T^{2} \) |
| 3 | \( 1 + 2.19T + 3T^{2} \) |
| 5 | \( 1 - 1.01T + 5T^{2} \) |
| 7 | \( 1 + 2.59T + 7T^{2} \) |
| 13 | \( 1 + 5.69T + 13T^{2} \) |
| 17 | \( 1 + 6.20T + 17T^{2} \) |
| 19 | \( 1 + 1.88T + 19T^{2} \) |
| 23 | \( 1 - 0.355T + 23T^{2} \) |
| 29 | \( 1 - 8.05T + 29T^{2} \) |
| 31 | \( 1 + 4.59T + 31T^{2} \) |
| 37 | \( 1 - 1.46T + 37T^{2} \) |
| 41 | \( 1 + 10.0T + 41T^{2} \) |
| 43 | \( 1 + 3.47T + 43T^{2} \) |
| 47 | \( 1 - 6.40T + 47T^{2} \) |
| 53 | \( 1 + 6.52T + 53T^{2} \) |
| 59 | \( 1 - 2.56T + 59T^{2} \) |
| 61 | \( 1 + 14.2T + 61T^{2} \) |
| 67 | \( 1 + 6.19T + 67T^{2} \) |
| 71 | \( 1 - 12.2T + 71T^{2} \) |
| 73 | \( 1 - 13.3T + 73T^{2} \) |
| 79 | \( 1 + 2.33T + 79T^{2} \) |
| 83 | \( 1 + 3.85T + 83T^{2} \) |
| 89 | \( 1 + 12.7T + 89T^{2} \) |
| 97 | \( 1 + 5.58T + 97T^{2} \) |
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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
−9.661955653529192224131796169522, −8.951262710848967411154152096440, −8.013696498049550588934096993180, −6.91525863656117644868327012483, −6.63874363737773200271110340960, −5.83251897495150725984651704504, −4.76171217554986434861015914247, −2.87373302413468269965290363217, −1.95847264112232902830613961140, −0.29734302983871377798042910694,
0.29734302983871377798042910694, 1.95847264112232902830613961140, 2.87373302413468269965290363217, 4.76171217554986434861015914247, 5.83251897495150725984651704504, 6.63874363737773200271110340960, 6.91525863656117644868327012483, 8.013696498049550588934096993180, 8.951262710848967411154152096440, 9.661955653529192224131796169522
