| L(s) = 1 | − 2.30·2-s − 3-s + 3.29·4-s − 1.89·5-s + 2.30·6-s − 2.97·8-s + 9-s + 4.36·10-s + 2.74·11-s − 3.29·12-s + 2.52·13-s + 1.89·15-s + 0.257·16-s − 1.10·17-s − 2.30·18-s − 5.86·19-s − 6.25·20-s − 6.30·22-s + 2.16·23-s + 2.97·24-s − 1.39·25-s − 5.81·26-s − 27-s + 9.39·29-s − 4.36·30-s + 1.86·31-s + 5.35·32-s + ⋯ |
| L(s) = 1 | − 1.62·2-s − 0.577·3-s + 1.64·4-s − 0.849·5-s + 0.939·6-s − 1.05·8-s + 0.333·9-s + 1.38·10-s + 0.826·11-s − 0.950·12-s + 0.700·13-s + 0.490·15-s + 0.0644·16-s − 0.267·17-s − 0.542·18-s − 1.34·19-s − 1.39·20-s − 1.34·22-s + 0.451·23-s + 0.607·24-s − 0.278·25-s − 1.13·26-s − 0.192·27-s + 1.74·29-s − 0.797·30-s + 0.335·31-s + 0.946·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 7 | \( 1 \) |
| 41 | \( 1 - T \) |
| good | 2 | \( 1 + 2.30T + 2T^{2} \) |
| 5 | \( 1 + 1.89T + 5T^{2} \) |
| 11 | \( 1 - 2.74T + 11T^{2} \) |
| 13 | \( 1 - 2.52T + 13T^{2} \) |
| 17 | \( 1 + 1.10T + 17T^{2} \) |
| 19 | \( 1 + 5.86T + 19T^{2} \) |
| 23 | \( 1 - 2.16T + 23T^{2} \) |
| 29 | \( 1 - 9.39T + 29T^{2} \) |
| 31 | \( 1 - 1.86T + 31T^{2} \) |
| 37 | \( 1 + 1.49T + 37T^{2} \) |
| 43 | \( 1 + 5.48T + 43T^{2} \) |
| 47 | \( 1 + 1.12T + 47T^{2} \) |
| 53 | \( 1 + 2.06T + 53T^{2} \) |
| 59 | \( 1 - 5.50T + 59T^{2} \) |
| 61 | \( 1 + 9.63T + 61T^{2} \) |
| 67 | \( 1 + 7.60T + 67T^{2} \) |
| 71 | \( 1 - 3.91T + 71T^{2} \) |
| 73 | \( 1 + 2.53T + 73T^{2} \) |
| 79 | \( 1 - 2.06T + 79T^{2} \) |
| 83 | \( 1 + 17.7T + 83T^{2} \) |
| 89 | \( 1 + 9.34T + 89T^{2} \) |
| 97 | \( 1 + 13.4T + 97T^{2} \) |
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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
−7.911794031713109346380029395205, −7.08931164102992424703148754458, −6.57131410827157199590541120427, −6.02035511474325514121364466619, −4.66552361927819688642258323632, −4.13453840930588948154102285747, −3.04207655389958411689156324613, −1.84607349328444075552392676068, −0.988880262270492529501408885392, 0,
0.988880262270492529501408885392, 1.84607349328444075552392676068, 3.04207655389958411689156324613, 4.13453840930588948154102285747, 4.66552361927819688642258323632, 6.02035511474325514121364466619, 6.57131410827157199590541120427, 7.08931164102992424703148754458, 7.911794031713109346380029395205
