| L(s) = 1 | − 2-s − 2.82·3-s + 4-s − 3.52·5-s + 2.82·6-s + 3.82·7-s − 8-s + 4.97·9-s + 3.52·10-s + 0.327·11-s − 2.82·12-s − 0.108·13-s − 3.82·14-s + 9.94·15-s + 16-s − 7.83·17-s − 4.97·18-s + 2.78·19-s − 3.52·20-s − 10.8·21-s − 0.327·22-s + 23-s + 2.82·24-s + 7.40·25-s + 0.108·26-s − 5.56·27-s + 3.82·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.63·3-s + 0.5·4-s − 1.57·5-s + 1.15·6-s + 1.44·7-s − 0.353·8-s + 1.65·9-s + 1.11·10-s + 0.0987·11-s − 0.815·12-s − 0.0300·13-s − 1.02·14-s + 2.56·15-s + 0.250·16-s − 1.89·17-s − 1.17·18-s + 0.639·19-s − 0.787·20-s − 2.35·21-s − 0.0698·22-s + 0.208·23-s + 0.576·24-s + 1.48·25-s + 0.0212·26-s − 1.07·27-s + 0.723·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5038967021\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5038967021\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| 131 | \( 1 + T \) |
| good | 3 | \( 1 + 2.82T + 3T^{2} \) |
| 5 | \( 1 + 3.52T + 5T^{2} \) |
| 7 | \( 1 - 3.82T + 7T^{2} \) |
| 11 | \( 1 - 0.327T + 11T^{2} \) |
| 13 | \( 1 + 0.108T + 13T^{2} \) |
| 17 | \( 1 + 7.83T + 17T^{2} \) |
| 19 | \( 1 - 2.78T + 19T^{2} \) |
| 29 | \( 1 - 7.73T + 29T^{2} \) |
| 31 | \( 1 + 3.33T + 31T^{2} \) |
| 37 | \( 1 - 5.79T + 37T^{2} \) |
| 41 | \( 1 - 8.97T + 41T^{2} \) |
| 43 | \( 1 - 6.14T + 43T^{2} \) |
| 47 | \( 1 + 7.13T + 47T^{2} \) |
| 53 | \( 1 + 1.61T + 53T^{2} \) |
| 59 | \( 1 + 6.37T + 59T^{2} \) |
| 61 | \( 1 - 14.3T + 61T^{2} \) |
| 67 | \( 1 + 2.59T + 67T^{2} \) |
| 71 | \( 1 - 6.48T + 71T^{2} \) |
| 73 | \( 1 + 12.2T + 73T^{2} \) |
| 79 | \( 1 - 15.3T + 79T^{2} \) |
| 83 | \( 1 + 4.17T + 83T^{2} \) |
| 89 | \( 1 + 15.5T + 89T^{2} \) |
| 97 | \( 1 + 14.4T + 97T^{2} \) |
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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
−8.025144242909736208670326128918, −7.35409159731041491825984602368, −6.82776130594683624476999051567, −6.05969034549608060906951585696, −5.08220778020577965999839532054, −4.54722322266812908202961619966, −4.03683851528021565600756125479, −2.58420311578701654674280519508, −1.31833915766921458660106380203, −0.50927231842327334781328640739,
0.50927231842327334781328640739, 1.31833915766921458660106380203, 2.58420311578701654674280519508, 4.03683851528021565600756125479, 4.54722322266812908202961619966, 5.08220778020577965999839532054, 6.05969034549608060906951585696, 6.82776130594683624476999051567, 7.35409159731041491825984602368, 8.025144242909736208670326128918
