| L(s) = 1 | − 2-s + 4-s − 8-s − 6·11-s − 13-s + 16-s + 2·17-s + 4·19-s + 6·22-s − 6·23-s + 26-s − 2·29-s − 4·31-s − 32-s − 2·34-s + 6·37-s − 4·38-s − 2·41-s + 6·43-s − 6·44-s + 6·46-s − 10·47-s − 52-s + 2·58-s − 12·59-s + 4·62-s + 64-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.353·8-s − 1.80·11-s − 0.277·13-s + 1/4·16-s + 0.485·17-s + 0.917·19-s + 1.27·22-s − 1.25·23-s + 0.196·26-s − 0.371·29-s − 0.718·31-s − 0.176·32-s − 0.342·34-s + 0.986·37-s − 0.648·38-s − 0.312·41-s + 0.914·43-s − 0.904·44-s + 0.884·46-s − 1.45·47-s − 0.138·52-s + 0.262·58-s − 1.56·59-s + 0.508·62-s + 1/8·64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 286650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 286650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3526571491\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3526571491\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 11 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 + 10 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 16 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 - 4 T + p T^{2} \) |
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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
−12.58156272157336, −12.40357517239342, −11.65532677734481, −11.27666891124834, −10.89181269841016, −10.21555317024634, −10.08259184980364, −9.536999985691656, −9.159303294774014, −8.449889931679254, −7.915840180688119, −7.745041090942633, −7.403660572503892, −6.673732418462188, −6.139846693570046, −5.544981757519956, −5.294290780201189, −4.685881265775493, −3.984392949308889, −3.342597274689786, −2.833235649032338, −2.347203867729264, −1.758048907683937, −1.056593837620459, −0.1901900631563117,
0.1901900631563117, 1.056593837620459, 1.758048907683937, 2.347203867729264, 2.833235649032338, 3.342597274689786, 3.984392949308889, 4.685881265775493, 5.294290780201189, 5.544981757519956, 6.139846693570046, 6.673732418462188, 7.403660572503892, 7.745041090942633, 7.915840180688119, 8.449889931679254, 9.159303294774014, 9.536999985691656, 10.08259184980364, 10.21555317024634, 10.89181269841016, 11.27666891124834, 11.65532677734481, 12.40357517239342, 12.58156272157336
