| L(s) = 1 | + 2.33·3-s − 3.77·7-s + 2.44·9-s + 3.77·11-s + 3.17·13-s − 1.39·17-s − 3.91·19-s − 8.80·21-s − 0.891·23-s − 1.30·27-s + 0.0492·29-s + 5.58·31-s + 8.80·33-s + 7.04·37-s + 7.40·39-s − 1.48·41-s − 2.69·43-s − 3.77·47-s + 7.24·49-s − 3.24·51-s + 11.6·53-s − 9.13·57-s + 0.690·59-s + 10.3·61-s − 9.21·63-s − 15.3·67-s − 2.07·69-s + ⋯ |
| L(s) = 1 | + 1.34·3-s − 1.42·7-s + 0.813·9-s + 1.13·11-s + 0.880·13-s − 0.337·17-s − 0.897·19-s − 1.92·21-s − 0.185·23-s − 0.250·27-s + 0.00913·29-s + 1.00·31-s + 1.53·33-s + 1.15·37-s + 1.18·39-s − 0.232·41-s − 0.411·43-s − 0.550·47-s + 1.03·49-s − 0.454·51-s + 1.59·53-s − 1.20·57-s + 0.0898·59-s + 1.32·61-s − 1.16·63-s − 1.87·67-s − 0.250·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10000 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.078642430\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.078642430\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 - 2.33T + 3T^{2} \) |
| 7 | \( 1 + 3.77T + 7T^{2} \) |
| 11 | \( 1 - 3.77T + 11T^{2} \) |
| 13 | \( 1 - 3.17T + 13T^{2} \) |
| 17 | \( 1 + 1.39T + 17T^{2} \) |
| 19 | \( 1 + 3.91T + 19T^{2} \) |
| 23 | \( 1 + 0.891T + 23T^{2} \) |
| 29 | \( 1 - 0.0492T + 29T^{2} \) |
| 31 | \( 1 - 5.58T + 31T^{2} \) |
| 37 | \( 1 - 7.04T + 37T^{2} \) |
| 41 | \( 1 + 1.48T + 41T^{2} \) |
| 43 | \( 1 + 2.69T + 43T^{2} \) |
| 47 | \( 1 + 3.77T + 47T^{2} \) |
| 53 | \( 1 - 11.6T + 53T^{2} \) |
| 59 | \( 1 - 0.690T + 59T^{2} \) |
| 61 | \( 1 - 10.3T + 61T^{2} \) |
| 67 | \( 1 + 15.3T + 67T^{2} \) |
| 71 | \( 1 - 6.69T + 71T^{2} \) |
| 73 | \( 1 - 5.16T + 73T^{2} \) |
| 79 | \( 1 - 1.80T + 79T^{2} \) |
| 83 | \( 1 - 9.96T + 83T^{2} \) |
| 89 | \( 1 - 14.5T + 89T^{2} \) |
| 97 | \( 1 + 0.0901T + 97T^{2} \) |
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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
−7.77423997538789058899099068462, −6.89535229760290946738977155384, −6.38529282827431718560417878858, −5.94285734691480153589672795095, −4.61076355711696200509536817426, −3.81310298541668999156103256090, −3.50801806508388689732079728274, −2.67648484461570621730461268619, −1.95287308010852908515455000427, −0.76715688754976897341615706285,
0.76715688754976897341615706285, 1.95287308010852908515455000427, 2.67648484461570621730461268619, 3.50801806508388689732079728274, 3.81310298541668999156103256090, 4.61076355711696200509536817426, 5.94285734691480153589672795095, 6.38529282827431718560417878858, 6.89535229760290946738977155384, 7.77423997538789058899099068462
