| L(s) = 1 | + 2-s + 3.07·3-s + 4-s + 3.99·5-s + 3.07·6-s − 0.788·7-s + 8-s + 6.44·9-s + 3.99·10-s + 2.99·11-s + 3.07·12-s − 4.03·13-s − 0.788·14-s + 12.2·15-s + 16-s − 6.47·17-s + 6.44·18-s + 1.44·19-s + 3.99·20-s − 2.42·21-s + 2.99·22-s − 0.160·23-s + 3.07·24-s + 10.9·25-s − 4.03·26-s + 10.5·27-s − 0.788·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.77·3-s + 0.5·4-s + 1.78·5-s + 1.25·6-s − 0.297·7-s + 0.353·8-s + 2.14·9-s + 1.26·10-s + 0.903·11-s + 0.887·12-s − 1.11·13-s − 0.210·14-s + 3.17·15-s + 0.250·16-s − 1.57·17-s + 1.51·18-s + 0.330·19-s + 0.893·20-s − 0.528·21-s + 0.638·22-s − 0.0334·23-s + 0.627·24-s + 2.19·25-s − 0.791·26-s + 2.03·27-s − 0.148·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(8.514855745\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.514855745\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 31 | \( 1 - T \) |
| 97 | \( 1 - T \) |
| good | 3 | \( 1 - 3.07T + 3T^{2} \) |
| 5 | \( 1 - 3.99T + 5T^{2} \) |
| 7 | \( 1 + 0.788T + 7T^{2} \) |
| 11 | \( 1 - 2.99T + 11T^{2} \) |
| 13 | \( 1 + 4.03T + 13T^{2} \) |
| 17 | \( 1 + 6.47T + 17T^{2} \) |
| 19 | \( 1 - 1.44T + 19T^{2} \) |
| 23 | \( 1 + 0.160T + 23T^{2} \) |
| 29 | \( 1 - 8.22T + 29T^{2} \) |
| 37 | \( 1 + 10.6T + 37T^{2} \) |
| 41 | \( 1 - 7.18T + 41T^{2} \) |
| 43 | \( 1 + 10.4T + 43T^{2} \) |
| 47 | \( 1 + 7.88T + 47T^{2} \) |
| 53 | \( 1 + 6.00T + 53T^{2} \) |
| 59 | \( 1 - 1.12T + 59T^{2} \) |
| 61 | \( 1 + 3.47T + 61T^{2} \) |
| 67 | \( 1 - 5.93T + 67T^{2} \) |
| 71 | \( 1 - 7.80T + 71T^{2} \) |
| 73 | \( 1 + 4.42T + 73T^{2} \) |
| 79 | \( 1 + 10.3T + 79T^{2} \) |
| 83 | \( 1 + 12.3T + 83T^{2} \) |
| 89 | \( 1 + 6.41T + 89T^{2} \) |
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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
−8.302759707270009731721481119318, −7.03874059426266780551114611277, −6.80008439076711701043204220258, −6.08633577866860650439835732697, −4.92666112776416177816291394029, −4.51232122047442326486378251747, −3.35865423880607263937692362279, −2.78713840368088809439403727086, −2.07055280090734079529018738028, −1.55070760675144550807014275779,
1.55070760675144550807014275779, 2.07055280090734079529018738028, 2.78713840368088809439403727086, 3.35865423880607263937692362279, 4.51232122047442326486378251747, 4.92666112776416177816291394029, 6.08633577866860650439835732697, 6.80008439076711701043204220258, 7.03874059426266780551114611277, 8.302759707270009731721481119318
