L(s) = 1 | − 3.14·3-s + 5-s − 3.14·7-s + 6.86·9-s − 7.00·13-s − 3.14·15-s + 5.14·17-s − 0.414·19-s + 9.86·21-s − 2.72·23-s + 25-s − 12.1·27-s + 5.86·29-s − 9.86·31-s − 3.14·35-s − 3.86·37-s + 22.0·39-s − 6.28·41-s + 6.72·43-s + 6.86·45-s − 9.00·47-s + 2.86·49-s − 16.1·51-s − 3.86·53-s + 1.30·57-s + 1.45·59-s + 10.6·61-s + ⋯ |
L(s) = 1 | − 1.81·3-s + 0.447·5-s − 1.18·7-s + 2.28·9-s − 1.94·13-s − 0.811·15-s + 1.24·17-s − 0.0951·19-s + 2.15·21-s − 0.568·23-s + 0.200·25-s − 2.33·27-s + 1.08·29-s − 1.77·31-s − 0.530·35-s − 0.635·37-s + 3.52·39-s − 0.981·41-s + 1.02·43-s + 1.02·45-s − 1.31·47-s + 0.409·49-s − 2.26·51-s − 0.531·53-s + 0.172·57-s + 0.189·59-s + 1.36·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4083549028\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4083549028\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 \) |
good | 3 | \( 1 + 3.14T + 3T^{2} \) |
| 7 | \( 1 + 3.14T + 7T^{2} \) |
| 13 | \( 1 + 7.00T + 13T^{2} \) |
| 17 | \( 1 - 5.14T + 17T^{2} \) |
| 19 | \( 1 + 0.414T + 19T^{2} \) |
| 23 | \( 1 + 2.72T + 23T^{2} \) |
| 29 | \( 1 - 5.86T + 29T^{2} \) |
| 31 | \( 1 + 9.86T + 31T^{2} \) |
| 37 | \( 1 + 3.86T + 37T^{2} \) |
| 41 | \( 1 + 6.28T + 41T^{2} \) |
| 43 | \( 1 - 6.72T + 43T^{2} \) |
| 47 | \( 1 + 9.00T + 47T^{2} \) |
| 53 | \( 1 + 3.86T + 53T^{2} \) |
| 59 | \( 1 - 1.45T + 59T^{2} \) |
| 61 | \( 1 - 10.6T + 61T^{2} \) |
| 67 | \( 1 - 7.29T + 67T^{2} \) |
| 71 | \( 1 + 6.69T + 71T^{2} \) |
| 73 | \( 1 + 8.72T + 73T^{2} \) |
| 79 | \( 1 - 12T + 79T^{2} \) |
| 83 | \( 1 + 4.17T + 83T^{2} \) |
| 89 | \( 1 + 16.4T + 89T^{2} \) |
| 97 | \( 1 - 6T + 97T^{2} \) |
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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
−8.078667687315795098531406652425, −7.07062443922606054773999835769, −6.87222441427386248992585788966, −5.99025267337360975318253354586, −5.40432573085708568751504697625, −4.92950860504219057940900747911, −3.91599166394901374814750864615, −2.87231592728251651165437308048, −1.67770646336179873720831323873, −0.38042757493823765793182904637,
0.38042757493823765793182904637, 1.67770646336179873720831323873, 2.87231592728251651165437308048, 3.91599166394901374814750864615, 4.92950860504219057940900747911, 5.40432573085708568751504697625, 5.99025267337360975318253354586, 6.87222441427386248992585788966, 7.07062443922606054773999835769, 8.078667687315795098531406652425