L(s) = 1 | + 3-s + 0.647·7-s + 9-s + 0.502·11-s + 13-s − 4.73·17-s − 6.58·19-s + 0.647·21-s + 2.85·23-s + 27-s + 0.294·29-s + 2.20·31-s + 0.502·33-s + 7.28·37-s + 39-s + 7.58·41-s + 0.444·43-s + 8.07·47-s − 6.58·49-s − 4.73·51-s − 12.7·53-s − 6.58·57-s + 10.8·59-s + 11.4·61-s + 0.647·63-s − 8.63·67-s + 2.85·69-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 0.244·7-s + 0.333·9-s + 0.151·11-s + 0.277·13-s − 1.14·17-s − 1.50·19-s + 0.141·21-s + 0.594·23-s + 0.192·27-s + 0.0547·29-s + 0.395·31-s + 0.0874·33-s + 1.19·37-s + 0.160·39-s + 1.18·41-s + 0.0678·43-s + 1.17·47-s − 0.940·49-s − 0.662·51-s − 1.74·53-s − 0.871·57-s + 1.40·59-s + 1.46·61-s + 0.0815·63-s − 1.05·67-s + 0.343·69-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
\(L(1)\) |
\(\approx\) |
\(2.530579890\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.530579890\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - 0.647T + 7T^{2} \) |
| 11 | \( 1 - 0.502T + 11T^{2} \) |
| 17 | \( 1 + 4.73T + 17T^{2} \) |
| 19 | \( 1 + 6.58T + 19T^{2} \) |
| 23 | \( 1 - 2.85T + 23T^{2} \) |
| 29 | \( 1 - 0.294T + 29T^{2} \) |
| 31 | \( 1 - 2.20T + 31T^{2} \) |
| 37 | \( 1 - 7.28T + 37T^{2} \) |
| 41 | \( 1 - 7.58T + 41T^{2} \) |
| 43 | \( 1 - 0.444T + 43T^{2} \) |
| 47 | \( 1 - 8.07T + 47T^{2} \) |
| 53 | \( 1 + 12.7T + 53T^{2} \) |
| 59 | \( 1 - 10.8T + 59T^{2} \) |
| 61 | \( 1 - 11.4T + 61T^{2} \) |
| 67 | \( 1 + 8.63T + 67T^{2} \) |
| 71 | \( 1 + 2.58T + 71T^{2} \) |
| 73 | \( 1 - 10.6T + 73T^{2} \) |
| 79 | \( 1 - 5.28T + 79T^{2} \) |
| 83 | \( 1 - 5.35T + 83T^{2} \) |
| 89 | \( 1 - 4.99T + 89T^{2} \) |
| 97 | \( 1 + 1.29T + 97T^{2} \) |
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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
−7.981211741652367171201571364612, −7.18532473576192883868776550606, −6.47963741119584403173043300633, −5.95669268582009978377571815774, −4.77754857782582545401027721121, −4.35592543162041928351219042786, −3.55415638166617779235034034498, −2.54208834624490875512241101630, −1.98638976906454830002208137368, −0.76451537744967454698853416196,
0.76451537744967454698853416196, 1.98638976906454830002208137368, 2.54208834624490875512241101630, 3.55415638166617779235034034498, 4.35592543162041928351219042786, 4.77754857782582545401027721121, 5.95669268582009978377571815774, 6.47963741119584403173043300633, 7.18532473576192883868776550606, 7.981211741652367171201571364612