| L(s) = 1 | − 0.792·2-s − 3-s − 1.37·4-s + 0.792·6-s − 2.52·7-s + 2.67·8-s + 9-s + 1.37·12-s − 6.78·13-s + 2·14-s + 0.627·16-s − 6.63·17-s − 0.792·18-s + 7.72·19-s + 2.52·21-s + 8·23-s − 2.67·24-s + 5.37·26-s − 27-s + 3.46·28-s + 3.16·29-s − 3.37·31-s − 5.84·32-s + 5.25·34-s − 1.37·36-s + 5·37-s − 6.11·38-s + ⋯ |
| L(s) = 1 | − 0.560·2-s − 0.577·3-s − 0.686·4-s + 0.323·6-s − 0.954·7-s + 0.944·8-s + 0.333·9-s + 0.396·12-s − 1.88·13-s + 0.534·14-s + 0.156·16-s − 1.60·17-s − 0.186·18-s + 1.77·19-s + 0.550·21-s + 1.66·23-s − 0.545·24-s + 1.05·26-s − 0.192·27-s + 0.654·28-s + 0.588·29-s − 0.605·31-s − 1.03·32-s + 0.901·34-s − 0.228·36-s + 0.821·37-s − 0.992·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9075 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9075 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3024022356\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3024022356\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 0.792T + 2T^{2} \) |
| 7 | \( 1 + 2.52T + 7T^{2} \) |
| 13 | \( 1 + 6.78T + 13T^{2} \) |
| 17 | \( 1 + 6.63T + 17T^{2} \) |
| 19 | \( 1 - 7.72T + 19T^{2} \) |
| 23 | \( 1 - 8T + 23T^{2} \) |
| 29 | \( 1 - 3.16T + 29T^{2} \) |
| 31 | \( 1 + 3.37T + 31T^{2} \) |
| 37 | \( 1 - 5T + 37T^{2} \) |
| 41 | \( 1 + 6.92T + 41T^{2} \) |
| 43 | \( 1 + 9.94T + 43T^{2} \) |
| 47 | \( 1 - 2.74T + 47T^{2} \) |
| 53 | \( 1 + 12.7T + 53T^{2} \) |
| 59 | \( 1 - 4T + 59T^{2} \) |
| 61 | \( 1 + 2.67T + 61T^{2} \) |
| 67 | \( 1 - 6.11T + 67T^{2} \) |
| 71 | \( 1 + 0.744T + 71T^{2} \) |
| 73 | \( 1 + 5.19T + 73T^{2} \) |
| 79 | \( 1 - 0.147T + 79T^{2} \) |
| 83 | \( 1 - 1.87T + 83T^{2} \) |
| 89 | \( 1 + 17.4T + 89T^{2} \) |
| 97 | \( 1 + 3.62T + 97T^{2} \) |
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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
−7.64463086674054385348181414628, −6.97313359155605649135322374470, −6.69645395729456967319265324049, −5.48369505572690581021023959685, −4.94886801440622726841952345873, −4.50333954128640190777380997918, −3.37930402626546639070943278359, −2.66578267326586907416951650500, −1.43197190776712929730520281370, −0.31297446947974179927242880302,
0.31297446947974179927242880302, 1.43197190776712929730520281370, 2.66578267326586907416951650500, 3.37930402626546639070943278359, 4.50333954128640190777380997918, 4.94886801440622726841952345873, 5.48369505572690581021023959685, 6.69645395729456967319265324049, 6.97313359155605649135322374470, 7.64463086674054385348181414628
