| L(s) = 1 | − 3.17·3-s + 5-s + 2.63·7-s + 7.04·9-s − 0.630·11-s + 5.95·13-s − 3.17·15-s + 3.26·17-s − 19-s − 8.34·21-s − 6.04·23-s + 25-s − 12.8·27-s − 8.34·29-s + 1.41·31-s + 2·33-s + 2.63·35-s − 6.29·37-s − 18.8·39-s − 11.1·43-s + 7.04·45-s + 4.78·47-s − 0.0783·49-s − 10.3·51-s − 11.3·53-s − 0.630·55-s + 3.17·57-s + ⋯ |
| L(s) = 1 | − 1.83·3-s + 0.447·5-s + 0.994·7-s + 2.34·9-s − 0.190·11-s + 1.65·13-s − 0.818·15-s + 0.791·17-s − 0.229·19-s − 1.81·21-s − 1.26·23-s + 0.200·25-s − 2.47·27-s − 1.54·29-s + 0.254·31-s + 0.348·33-s + 0.444·35-s − 1.03·37-s − 3.02·39-s − 1.69·43-s + 1.05·45-s + 0.698·47-s − 0.0111·49-s − 1.44·51-s − 1.56·53-s − 0.0850·55-s + 0.419·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 + 3.17T + 3T^{2} \) |
| 7 | \( 1 - 2.63T + 7T^{2} \) |
| 11 | \( 1 + 0.630T + 11T^{2} \) |
| 13 | \( 1 - 5.95T + 13T^{2} \) |
| 17 | \( 1 - 3.26T + 17T^{2} \) |
| 23 | \( 1 + 6.04T + 23T^{2} \) |
| 29 | \( 1 + 8.34T + 29T^{2} \) |
| 31 | \( 1 - 1.41T + 31T^{2} \) |
| 37 | \( 1 + 6.29T + 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 + 11.1T + 43T^{2} \) |
| 47 | \( 1 - 4.78T + 47T^{2} \) |
| 53 | \( 1 + 11.3T + 53T^{2} \) |
| 59 | \( 1 + 0.241T + 59T^{2} \) |
| 61 | \( 1 + 4.63T + 61T^{2} \) |
| 67 | \( 1 + 0.489T + 67T^{2} \) |
| 71 | \( 1 + 9.41T + 71T^{2} \) |
| 73 | \( 1 + 2.18T + 73T^{2} \) |
| 79 | \( 1 + 13.9T + 79T^{2} \) |
| 83 | \( 1 - 1.86T + 83T^{2} \) |
| 89 | \( 1 - 12.3T + 89T^{2} \) |
| 97 | \( 1 + 17.7T + 97T^{2} \) |
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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
−7.65554815460266497060807079875, −6.76839799407672564046715266769, −6.06301266009962951244141751708, −5.67728678571259893151447346832, −5.04726402243819073332958767627, −4.28135466929350083098218114177, −3.47959744372796817045915638482, −1.73665890103704599105035677211, −1.37440464071925722729212956652, 0,
1.37440464071925722729212956652, 1.73665890103704599105035677211, 3.47959744372796817045915638482, 4.28135466929350083098218114177, 5.04726402243819073332958767627, 5.67728678571259893151447346832, 6.06301266009962951244141751708, 6.76839799407672564046715266769, 7.65554815460266497060807079875
