| L(s) = 1 | − 3-s − 7-s + 9-s − 5.41·11-s + 4.34·13-s + 1.07·17-s − 4.34·19-s + 21-s − 6.34·23-s − 27-s − 8.83·29-s + 4.34·31-s + 5.41·33-s − 8.68·37-s − 4.34·39-s + 8.34·41-s − 6.15·43-s + 6.83·47-s + 49-s − 1.07·51-s + 6.18·53-s + 4.34·57-s + 6.83·59-s − 4.52·61-s − 63-s + 6.34·69-s + 14.0·71-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.377·7-s + 0.333·9-s − 1.63·11-s + 1.20·13-s + 0.261·17-s − 0.995·19-s + 0.218·21-s − 1.32·23-s − 0.192·27-s − 1.64·29-s + 0.779·31-s + 0.943·33-s − 1.42·37-s − 0.694·39-s + 1.30·41-s − 0.938·43-s + 0.997·47-s + 0.142·49-s − 0.151·51-s + 0.849·53-s + 0.574·57-s + 0.890·59-s − 0.579·61-s − 0.125·63-s + 0.763·69-s + 1.67·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8581020562\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8581020562\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 11 | \( 1 + 5.41T + 11T^{2} \) |
| 13 | \( 1 - 4.34T + 13T^{2} \) |
| 17 | \( 1 - 1.07T + 17T^{2} \) |
| 19 | \( 1 + 4.34T + 19T^{2} \) |
| 23 | \( 1 + 6.34T + 23T^{2} \) |
| 29 | \( 1 + 8.83T + 29T^{2} \) |
| 31 | \( 1 - 4.34T + 31T^{2} \) |
| 37 | \( 1 + 8.68T + 37T^{2} \) |
| 41 | \( 1 - 8.34T + 41T^{2} \) |
| 43 | \( 1 + 6.15T + 43T^{2} \) |
| 47 | \( 1 - 6.83T + 47T^{2} \) |
| 53 | \( 1 - 6.18T + 53T^{2} \) |
| 59 | \( 1 - 6.83T + 59T^{2} \) |
| 61 | \( 1 + 4.52T + 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 - 14.0T + 71T^{2} \) |
| 73 | \( 1 + 11.1T + 73T^{2} \) |
| 79 | \( 1 + 0.680T + 79T^{2} \) |
| 83 | \( 1 - 6.83T + 83T^{2} \) |
| 89 | \( 1 - 6.49T + 89T^{2} \) |
| 97 | \( 1 + 10.4T + 97T^{2} \) |
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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
−7.85209432911951650163840398209, −7.06398339613357458168138953429, −6.28964707019974493132793197468, −5.70917907657490054648972227718, −5.23348397665148064140240375067, −4.18274900224986497455678749070, −3.64086493167821816867331059024, −2.58528366037530987558107251950, −1.77737058994670304910461619279, −0.45276716010462051671441067729,
0.45276716010462051671441067729, 1.77737058994670304910461619279, 2.58528366037530987558107251950, 3.64086493167821816867331059024, 4.18274900224986497455678749070, 5.23348397665148064140240375067, 5.70917907657490054648972227718, 6.28964707019974493132793197468, 7.06398339613357458168138953429, 7.85209432911951650163840398209
