L(s) = 1 | + 3-s + 1.48·7-s + 9-s + 5.77·11-s − 13-s + 3·17-s + 2.29·19-s + 1.48·21-s − 4.29·23-s + 27-s + 6.55·29-s + 5.77·31-s + 5.77·33-s + 3.25·37-s − 39-s + 5.25·41-s + 5.25·43-s − 5.03·47-s − 4.80·49-s + 3·51-s − 3.29·53-s + 2.29·57-s − 5.03·59-s − 4.55·61-s + 1.48·63-s + 4.22·67-s − 4.29·69-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 0.559·7-s + 0.333·9-s + 1.74·11-s − 0.277·13-s + 0.727·17-s + 0.526·19-s + 0.323·21-s − 0.895·23-s + 0.192·27-s + 1.21·29-s + 1.03·31-s + 1.00·33-s + 0.535·37-s − 0.160·39-s + 0.820·41-s + 0.801·43-s − 0.734·47-s − 0.686·49-s + 0.420·51-s − 0.452·53-s + 0.304·57-s − 0.655·59-s − 0.582·61-s + 0.186·63-s + 0.516·67-s − 0.517·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.480700640\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.480700640\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 13 | \( 1 + T \) |
good | 7 | \( 1 - 1.48T + 7T^{2} \) |
| 11 | \( 1 - 5.77T + 11T^{2} \) |
| 17 | \( 1 - 3T + 17T^{2} \) |
| 19 | \( 1 - 2.29T + 19T^{2} \) |
| 23 | \( 1 + 4.29T + 23T^{2} \) |
| 29 | \( 1 - 6.55T + 29T^{2} \) |
| 31 | \( 1 - 5.77T + 31T^{2} \) |
| 37 | \( 1 - 3.25T + 37T^{2} \) |
| 41 | \( 1 - 5.25T + 41T^{2} \) |
| 43 | \( 1 - 5.25T + 43T^{2} \) |
| 47 | \( 1 + 5.03T + 47T^{2} \) |
| 53 | \( 1 + 3.29T + 53T^{2} \) |
| 59 | \( 1 + 5.03T + 59T^{2} \) |
| 61 | \( 1 + 4.55T + 61T^{2} \) |
| 67 | \( 1 - 4.22T + 67T^{2} \) |
| 71 | \( 1 + 12.8T + 71T^{2} \) |
| 73 | \( 1 + 12.8T + 73T^{2} \) |
| 79 | \( 1 + 1.25T + 79T^{2} \) |
| 83 | \( 1 + 12.0T + 83T^{2} \) |
| 89 | \( 1 - 13.5T + 89T^{2} \) |
| 97 | \( 1 - 6.96T + 97T^{2} \) |
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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
−7.77428662783601002419103982829, −7.38748860494862090270904796144, −6.35461242470062830841247372333, −6.01687707204312962909138970265, −4.77684653503503663264839759301, −4.35388704016881001649707589494, −3.49732200851230148692659161025, −2.75190589013385557680916195905, −1.67436161630326493790896512940, −0.998398824874256283170876879894,
0.998398824874256283170876879894, 1.67436161630326493790896512940, 2.75190589013385557680916195905, 3.49732200851230148692659161025, 4.35388704016881001649707589494, 4.77684653503503663264839759301, 6.01687707204312962909138970265, 6.35461242470062830841247372333, 7.38748860494862090270904796144, 7.77428662783601002419103982829