| L(s) = 1 | + 3-s − 5-s + 4.22·7-s + 9-s + 1.28·11-s − 13-s − 15-s − 0.945·17-s + 4.22·19-s + 4.22·21-s − 6.94·23-s + 25-s + 27-s − 4.94·29-s + 7.17·31-s + 1.28·33-s − 4.22·35-s + 7.89·37-s − 39-s − 10.4·41-s + 8·43-s − 45-s + 9.74·47-s + 10.8·49-s − 0.945·51-s − 6.45·53-s − 1.28·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s + 1.59·7-s + 0.333·9-s + 0.387·11-s − 0.277·13-s − 0.258·15-s − 0.229·17-s + 0.970·19-s + 0.923·21-s − 1.44·23-s + 0.200·25-s + 0.192·27-s − 0.918·29-s + 1.28·31-s + 0.223·33-s − 0.714·35-s + 1.29·37-s − 0.160·39-s − 1.63·41-s + 1.21·43-s − 0.149·45-s + 1.42·47-s + 1.55·49-s − 0.132·51-s − 0.887·53-s − 0.173·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.004874022\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.004874022\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| good | 7 | \( 1 - 4.22T + 7T^{2} \) |
| 11 | \( 1 - 1.28T + 11T^{2} \) |
| 17 | \( 1 + 0.945T + 17T^{2} \) |
| 19 | \( 1 - 4.22T + 19T^{2} \) |
| 23 | \( 1 + 6.94T + 23T^{2} \) |
| 29 | \( 1 + 4.94T + 29T^{2} \) |
| 31 | \( 1 - 7.17T + 31T^{2} \) |
| 37 | \( 1 - 7.89T + 37T^{2} \) |
| 41 | \( 1 + 10.4T + 41T^{2} \) |
| 43 | \( 1 - 8T + 43T^{2} \) |
| 47 | \( 1 - 9.74T + 47T^{2} \) |
| 53 | \( 1 + 6.45T + 53T^{2} \) |
| 59 | \( 1 - 9.28T + 59T^{2} \) |
| 61 | \( 1 - 6T + 61T^{2} \) |
| 67 | \( 1 - 11.1T + 67T^{2} \) |
| 71 | \( 1 - 8.60T + 71T^{2} \) |
| 73 | \( 1 - 0.945T + 73T^{2} \) |
| 79 | \( 1 + 14.3T + 79T^{2} \) |
| 83 | \( 1 - 4.60T + 83T^{2} \) |
| 89 | \( 1 - 2T + 89T^{2} \) |
| 97 | \( 1 + 10.8T + 97T^{2} \) |
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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
−8.158936648403758090876833876530, −7.52442496233376003678886569346, −6.86757150490645623483068523862, −5.80779530955460470308978885352, −5.08493754991310688738317248976, −4.28806138662947114414926780729, −3.82252028372654627267485948786, −2.64859702717838080027124488165, −1.88942843508374005551163601488, −0.924500426027241251214556309842,
0.924500426027241251214556309842, 1.88942843508374005551163601488, 2.64859702717838080027124488165, 3.82252028372654627267485948786, 4.28806138662947114414926780729, 5.08493754991310688738317248976, 5.80779530955460470308978885352, 6.86757150490645623483068523862, 7.52442496233376003678886569346, 8.158936648403758090876833876530
