| L(s) = 1 | + 3.33·2-s + 3.14·4-s − 7·7-s − 16.2·8-s − 18.3·11-s − 10.1·13-s − 23.3·14-s − 79.2·16-s − 24.6·17-s + 77.4·19-s − 61.1·22-s + 149.·23-s − 33.8·26-s − 21.9·28-s + 10.2·29-s + 124.·31-s − 134.·32-s − 82.4·34-s − 215.·37-s + 258.·38-s − 495.·41-s + 220.·43-s − 57.5·44-s + 498.·46-s + 212.·47-s + 49·49-s − 31.8·52-s + ⋯ |
| L(s) = 1 | + 1.18·2-s + 0.392·4-s − 0.377·7-s − 0.716·8-s − 0.502·11-s − 0.216·13-s − 0.446·14-s − 1.23·16-s − 0.352·17-s + 0.934·19-s − 0.592·22-s + 1.35·23-s − 0.255·26-s − 0.148·28-s + 0.0659·29-s + 0.720·31-s − 0.744·32-s − 0.415·34-s − 0.959·37-s + 1.10·38-s − 1.88·41-s + 0.782·43-s − 0.197·44-s + 1.59·46-s + 0.660·47-s + 0.142·49-s − 0.0849·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.009138437\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.009138437\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + 7T \) |
| good | 2 | \( 1 - 3.33T + 8T^{2} \) |
| 11 | \( 1 + 18.3T + 1.33e3T^{2} \) |
| 13 | \( 1 + 10.1T + 2.19e3T^{2} \) |
| 17 | \( 1 + 24.6T + 4.91e3T^{2} \) |
| 19 | \( 1 - 77.4T + 6.85e3T^{2} \) |
| 23 | \( 1 - 149.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 10.2T + 2.43e4T^{2} \) |
| 31 | \( 1 - 124.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 215.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 495.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 220.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 212.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 532.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 324.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 653.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 819.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 466.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 173.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 810.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 12.3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 33.8T + 7.04e5T^{2} \) |
| 97 | \( 1 - 810.T + 9.12e5T^{2} \) |
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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
−9.063566974122904023729936120641, −8.305917135510883729128618834862, −7.10679603257286591009806923793, −6.59092816570984299802052981936, −5.38112958523746262568731668368, −5.11890498743356912505225611711, −3.98724878811472988546225419792, −3.19426698097038342493957529102, −2.36588803059966364284598863479, −0.68468179045546496517568160268,
0.68468179045546496517568160268, 2.36588803059966364284598863479, 3.19426698097038342493957529102, 3.98724878811472988546225419792, 5.11890498743356912505225611711, 5.38112958523746262568731668368, 6.59092816570984299802052981936, 7.10679603257286591009806923793, 8.305917135510883729128618834862, 9.063566974122904023729936120641
