L(s) = 1 | + 2-s + 4-s + 7-s + 8-s − 1.73·11-s − 5.46·13-s + 14-s + 16-s + 4·17-s + 5.73·19-s − 1.73·22-s + 2.46·23-s − 5.46·26-s + 28-s − 1.46·29-s − 0.267·31-s + 32-s + 4·34-s + 3.19·37-s + 5.73·38-s + 3.92·41-s + 4.53·43-s − 1.73·44-s + 2.46·46-s + 11.4·47-s + 49-s − 5.46·52-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.5·4-s + 0.377·7-s + 0.353·8-s − 0.522·11-s − 1.51·13-s + 0.267·14-s + 0.250·16-s + 0.970·17-s + 1.31·19-s − 0.369·22-s + 0.513·23-s − 1.07·26-s + 0.188·28-s − 0.271·29-s − 0.0481·31-s + 0.176·32-s + 0.685·34-s + 0.525·37-s + 0.929·38-s + 0.613·41-s + 0.691·43-s − 0.261·44-s + 0.363·46-s + 1.67·47-s + 0.142·49-s − 0.757·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.413056448\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.413056448\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
good | 11 | \( 1 + 1.73T + 11T^{2} \) |
| 13 | \( 1 + 5.46T + 13T^{2} \) |
| 17 | \( 1 - 4T + 17T^{2} \) |
| 19 | \( 1 - 5.73T + 19T^{2} \) |
| 23 | \( 1 - 2.46T + 23T^{2} \) |
| 29 | \( 1 + 1.46T + 29T^{2} \) |
| 31 | \( 1 + 0.267T + 31T^{2} \) |
| 37 | \( 1 - 3.19T + 37T^{2} \) |
| 41 | \( 1 - 3.92T + 41T^{2} \) |
| 43 | \( 1 - 4.53T + 43T^{2} \) |
| 47 | \( 1 - 11.4T + 47T^{2} \) |
| 53 | \( 1 + 13.8T + 53T^{2} \) |
| 59 | \( 1 + 4.92T + 59T^{2} \) |
| 61 | \( 1 + 13.8T + 61T^{2} \) |
| 67 | \( 1 - 12.3T + 67T^{2} \) |
| 71 | \( 1 + 4.26T + 71T^{2} \) |
| 73 | \( 1 + 16.9T + 73T^{2} \) |
| 79 | \( 1 - 8.92T + 79T^{2} \) |
| 83 | \( 1 - 9.46T + 83T^{2} \) |
| 89 | \( 1 + T + 89T^{2} \) |
| 97 | \( 1 - 6.92T + 97T^{2} \) |
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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
−7.48547119668401162695854659721, −7.28206586646266552736610757147, −6.11424666223687944322368273447, −5.54626575236342589924684132457, −4.91194540177050100715608673949, −4.42252992477271427311391874327, −3.30074612997481487973740371713, −2.80928999505812828281901754044, −1.90922888309098944470021687333, −0.794369016615572623477208408104,
0.794369016615572623477208408104, 1.90922888309098944470021687333, 2.80928999505812828281901754044, 3.30074612997481487973740371713, 4.42252992477271427311391874327, 4.91194540177050100715608673949, 5.54626575236342589924684132457, 6.11424666223687944322368273447, 7.28206586646266552736610757147, 7.48547119668401162695854659721