L(s) = 1 | + 5-s − 4.41·7-s − 0.245·11-s + 0.725·13-s − 0.737·17-s + 1.75·19-s + 23-s + 25-s − 10.2·29-s − 3.46·31-s − 4.41·35-s + 0.273·37-s + 6.36·41-s + 1.02·43-s + 3.89·47-s + 12.5·49-s − 5.12·53-s − 0.245·55-s + 3.69·59-s − 9.32·61-s + 0.725·65-s + 8.56·67-s − 2.72·71-s + 13.2·73-s + 1.08·77-s − 13.0·79-s − 16.2·83-s + ⋯ |
L(s) = 1 | + 0.447·5-s − 1.66·7-s − 0.0740·11-s + 0.201·13-s − 0.178·17-s + 0.402·19-s + 0.208·23-s + 0.200·25-s − 1.89·29-s − 0.621·31-s − 0.746·35-s + 0.0449·37-s + 0.993·41-s + 0.156·43-s + 0.568·47-s + 1.78·49-s − 0.704·53-s − 0.0331·55-s + 0.480·59-s − 1.19·61-s + 0.0900·65-s + 1.04·67-s − 0.322·71-s + 1.55·73-s + 0.123·77-s − 1.47·79-s − 1.78·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.348098918\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.348098918\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 23 | \( 1 - T \) |
good | 7 | \( 1 + 4.41T + 7T^{2} \) |
| 11 | \( 1 + 0.245T + 11T^{2} \) |
| 13 | \( 1 - 0.725T + 13T^{2} \) |
| 17 | \( 1 + 0.737T + 17T^{2} \) |
| 19 | \( 1 - 1.75T + 19T^{2} \) |
| 29 | \( 1 + 10.2T + 29T^{2} \) |
| 31 | \( 1 + 3.46T + 31T^{2} \) |
| 37 | \( 1 - 0.273T + 37T^{2} \) |
| 41 | \( 1 - 6.36T + 41T^{2} \) |
| 43 | \( 1 - 1.02T + 43T^{2} \) |
| 47 | \( 1 - 3.89T + 47T^{2} \) |
| 53 | \( 1 + 5.12T + 53T^{2} \) |
| 59 | \( 1 - 3.69T + 59T^{2} \) |
| 61 | \( 1 + 9.32T + 61T^{2} \) |
| 67 | \( 1 - 8.56T + 67T^{2} \) |
| 71 | \( 1 + 2.72T + 71T^{2} \) |
| 73 | \( 1 - 13.2T + 73T^{2} \) |
| 79 | \( 1 + 13.0T + 79T^{2} \) |
| 83 | \( 1 + 16.2T + 83T^{2} \) |
| 89 | \( 1 - 14.1T + 89T^{2} \) |
| 97 | \( 1 - 18.6T + 97T^{2} \) |
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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
−7.57356804002162474255250513351, −7.14852608312616589903707621897, −6.30644332886207574820135050979, −5.88270530532947663223530817401, −5.19079379506193749077673212605, −4.09031111654381802645672123338, −3.45493870074482052559131606654, −2.75439491999202419768320496715, −1.83684426880189937497365115176, −0.55212206858907260519352753040,
0.55212206858907260519352753040, 1.83684426880189937497365115176, 2.75439491999202419768320496715, 3.45493870074482052559131606654, 4.09031111654381802645672123338, 5.19079379506193749077673212605, 5.88270530532947663223530817401, 6.30644332886207574820135050979, 7.14852608312616589903707621897, 7.57356804002162474255250513351