| L(s) = 1 | − 2·5-s + 4·11-s + 2·13-s − 6·17-s − 8·19-s − 25-s + 6·29-s + 8·31-s + 2·37-s + 2·41-s − 4·43-s + 8·47-s + 6·53-s − 8·55-s − 6·61-s − 4·65-s − 4·67-s − 8·71-s − 10·73-s − 16·79-s + 8·83-s + 12·85-s − 6·89-s + 16·95-s + 6·97-s + 101-s + 103-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 1.20·11-s + 0.554·13-s − 1.45·17-s − 1.83·19-s − 1/5·25-s + 1.11·29-s + 1.43·31-s + 0.328·37-s + 0.312·41-s − 0.609·43-s + 1.16·47-s + 0.824·53-s − 1.07·55-s − 0.768·61-s − 0.496·65-s − 0.488·67-s − 0.949·71-s − 1.17·73-s − 1.80·79-s + 0.878·83-s + 1.30·85-s − 0.635·89-s + 1.64·95-s + 0.609·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 28224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28224 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.417252480\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.417252480\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−15.29056941823743, −14.77959388599602, −14.22405260533457, −13.46461994337212, −13.26466633985677, −12.36966823091561, −11.98985527848339, −11.52266792663092, −10.94926418482203, −10.49478116255371, −9.827500787238704, −8.997855416465995, −8.531844150278744, −8.380916128111494, −7.392934534027812, −6.867823192044496, −6.258298249066854, −5.973746163245168, −4.643744690908617, −4.318274102378088, −3.981186380549427, −3.035535424252344, −2.296080728763474, −1.434388625461478, −0.4756283302840617,
0.4756283302840617, 1.434388625461478, 2.296080728763474, 3.035535424252344, 3.981186380549427, 4.318274102378088, 4.643744690908617, 5.973746163245168, 6.258298249066854, 6.867823192044496, 7.392934534027812, 8.380916128111494, 8.531844150278744, 8.997855416465995, 9.827500787238704, 10.49478116255371, 10.94926418482203, 11.52266792663092, 11.98985527848339, 12.36966823091561, 13.26466633985677, 13.46461994337212, 14.22405260533457, 14.77959388599602, 15.29056941823743
