| L(s) = 1 | + 3-s − 2.32·7-s + 9-s − 6.59·11-s + 3.56·13-s + 5.87·17-s − 5.15·19-s − 2.32·21-s − 2.77·23-s + 27-s − 9.73·29-s − 31-s − 6.59·33-s + 5.76·37-s + 3.56·39-s + 2.75·41-s + 8.84·43-s + 2.85·47-s − 1.60·49-s + 5.87·51-s − 10.7·53-s − 5.15·57-s + 2.98·59-s + 2.17·61-s − 2.32·63-s + 4.65·67-s − 2.77·69-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.877·7-s + 0.333·9-s − 1.98·11-s + 0.989·13-s + 1.42·17-s − 1.18·19-s − 0.506·21-s − 0.578·23-s + 0.192·27-s − 1.80·29-s − 0.179·31-s − 1.14·33-s + 0.947·37-s + 0.571·39-s + 0.430·41-s + 1.34·43-s + 0.416·47-s − 0.229·49-s + 0.821·51-s − 1.47·53-s − 0.683·57-s + 0.388·59-s + 0.278·61-s − 0.292·63-s + 0.568·67-s − 0.334·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.672075597\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.672075597\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 31 | \( 1 + T \) |
| good | 7 | \( 1 + 2.32T + 7T^{2} \) |
| 11 | \( 1 + 6.59T + 11T^{2} \) |
| 13 | \( 1 - 3.56T + 13T^{2} \) |
| 17 | \( 1 - 5.87T + 17T^{2} \) |
| 19 | \( 1 + 5.15T + 19T^{2} \) |
| 23 | \( 1 + 2.77T + 23T^{2} \) |
| 29 | \( 1 + 9.73T + 29T^{2} \) |
| 37 | \( 1 - 5.76T + 37T^{2} \) |
| 41 | \( 1 - 2.75T + 41T^{2} \) |
| 43 | \( 1 - 8.84T + 43T^{2} \) |
| 47 | \( 1 - 2.85T + 47T^{2} \) |
| 53 | \( 1 + 10.7T + 53T^{2} \) |
| 59 | \( 1 - 2.98T + 59T^{2} \) |
| 61 | \( 1 - 2.17T + 61T^{2} \) |
| 67 | \( 1 - 4.65T + 67T^{2} \) |
| 71 | \( 1 - 15.3T + 71T^{2} \) |
| 73 | \( 1 - 2T + 73T^{2} \) |
| 79 | \( 1 - 4.41T + 79T^{2} \) |
| 83 | \( 1 + 11.6T + 83T^{2} \) |
| 89 | \( 1 - 9.45T + 89T^{2} \) |
| 97 | \( 1 + 1.28T + 97T^{2} \) |
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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
−7.86610052648070875794655465934, −7.22924744891183722586935772851, −6.16741656668869763095781121118, −5.80917147238975656626135522329, −5.00671260575771510503759005380, −3.96858385505821964678269142101, −3.43231568737947989955234882871, −2.65938187451161194733725218834, −1.94779983667264461779402220147, −0.57590400178220054847895308641,
0.57590400178220054847895308641, 1.94779983667264461779402220147, 2.65938187451161194733725218834, 3.43231568737947989955234882871, 3.96858385505821964678269142101, 5.00671260575771510503759005380, 5.80917147238975656626135522329, 6.16741656668869763095781121118, 7.22924744891183722586935772851, 7.86610052648070875794655465934
