| L(s) = 1 | + 2-s + 2.22·3-s + 4-s + 3.08·5-s + 2.22·6-s + 7-s + 8-s + 1.96·9-s + 3.08·10-s + 1.14·11-s + 2.22·12-s − 3.25·13-s + 14-s + 6.87·15-s + 16-s + 4.21·17-s + 1.96·18-s + 3.08·20-s + 2.22·21-s + 1.14·22-s − 0.445·23-s + 2.22·24-s + 4.52·25-s − 3.25·26-s − 2.30·27-s + 28-s − 9.88·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.28·3-s + 0.5·4-s + 1.38·5-s + 0.909·6-s + 0.377·7-s + 0.353·8-s + 0.654·9-s + 0.976·10-s + 0.345·11-s + 0.643·12-s − 0.903·13-s + 0.267·14-s + 1.77·15-s + 0.250·16-s + 1.02·17-s + 0.462·18-s + 0.690·20-s + 0.486·21-s + 0.244·22-s − 0.0929·23-s + 0.454·24-s + 0.905·25-s − 0.639·26-s − 0.444·27-s + 0.188·28-s − 1.83·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5054 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5054 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.819937813\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.819937813\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 - 2.22T + 3T^{2} \) |
| 5 | \( 1 - 3.08T + 5T^{2} \) |
| 11 | \( 1 - 1.14T + 11T^{2} \) |
| 13 | \( 1 + 3.25T + 13T^{2} \) |
| 17 | \( 1 - 4.21T + 17T^{2} \) |
| 23 | \( 1 + 0.445T + 23T^{2} \) |
| 29 | \( 1 + 9.88T + 29T^{2} \) |
| 31 | \( 1 - 1.67T + 31T^{2} \) |
| 37 | \( 1 - 8.76T + 37T^{2} \) |
| 41 | \( 1 - 4.91T + 41T^{2} \) |
| 43 | \( 1 - 4.44T + 43T^{2} \) |
| 47 | \( 1 - 2.20T + 47T^{2} \) |
| 53 | \( 1 + 11.2T + 53T^{2} \) |
| 59 | \( 1 + 6.58T + 59T^{2} \) |
| 61 | \( 1 - 7.80T + 61T^{2} \) |
| 67 | \( 1 + 0.524T + 67T^{2} \) |
| 71 | \( 1 - 4.81T + 71T^{2} \) |
| 73 | \( 1 - 8.13T + 73T^{2} \) |
| 79 | \( 1 - 10.7T + 79T^{2} \) |
| 83 | \( 1 + 13.1T + 83T^{2} \) |
| 89 | \( 1 + 16.8T + 89T^{2} \) |
| 97 | \( 1 - 13.5T + 97T^{2} \) |
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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
−7.990307189842674929865753792861, −7.69530580402624176840095229109, −6.77680954866613224155642040855, −5.84353554205105654504987420601, −5.44106846211082356328644834417, −4.45399258682087081438724612355, −3.63134259965231073941765289650, −2.72333235765668763068639052072, −2.20882012673850199813727605203, −1.38888642801720435226278646263,
1.38888642801720435226278646263, 2.20882012673850199813727605203, 2.72333235765668763068639052072, 3.63134259965231073941765289650, 4.45399258682087081438724612355, 5.44106846211082356328644834417, 5.84353554205105654504987420601, 6.77680954866613224155642040855, 7.69530580402624176840095229109, 7.990307189842674929865753792861
