L(s) = 1 | + 5-s + 11-s − 6·13-s + 6·17-s + 4·19-s − 4·23-s + 25-s − 10·29-s + 4·31-s + 6·37-s + 10·43-s − 8·47-s + 6·53-s + 55-s + 8·59-s + 2·61-s − 6·65-s + 8·67-s − 6·71-s + 10·73-s + 2·79-s + 12·83-s + 6·85-s + 2·89-s + 4·95-s − 6·97-s + 101-s + ⋯ |
L(s) = 1 | + 0.447·5-s + 0.301·11-s − 1.66·13-s + 1.45·17-s + 0.917·19-s − 0.834·23-s + 1/5·25-s − 1.85·29-s + 0.718·31-s + 0.986·37-s + 1.52·43-s − 1.16·47-s + 0.824·53-s + 0.134·55-s + 1.04·59-s + 0.256·61-s − 0.744·65-s + 0.977·67-s − 0.712·71-s + 1.17·73-s + 0.225·79-s + 1.31·83-s + 0.650·85-s + 0.211·89-s + 0.410·95-s − 0.609·97-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 388080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 388080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.946948987\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.946948987\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
good | 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 2 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 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
−12.34048782248403, −12.11163557433084, −11.61610236801861, −11.19445128347283, −10.58216181139215, −10.00177239783643, −9.781380054459198, −9.393490003416766, −9.103314135242132, −8.102206242964130, −7.895372176068636, −7.545924090321546, −6.902317003068766, −6.589149340151674, −5.714589984475503, −5.536158916257947, −5.178231899626544, −4.435659520101262, −3.965987431622440, −3.410305035076036, −2.786116811600265, −2.327113023654359, −1.779235269616112, −1.043866505714062, −0.4802757291197156,
0.4802757291197156, 1.043866505714062, 1.779235269616112, 2.327113023654359, 2.786116811600265, 3.410305035076036, 3.965987431622440, 4.435659520101262, 5.178231899626544, 5.536158916257947, 5.714589984475503, 6.589149340151674, 6.902317003068766, 7.545924090321546, 7.895372176068636, 8.102206242964130, 9.103314135242132, 9.393490003416766, 9.781380054459198, 10.00177239783643, 10.58216181139215, 11.19445128347283, 11.61610236801861, 12.11163557433084, 12.34048782248403