L(s) = 1 | − 3·3-s − 5·5-s − 7·7-s + 9·9-s − 24.5·11-s − 35.0·13-s + 15·15-s − 18.4·17-s + 67.4·19-s + 21·21-s + 145.·23-s + 25·25-s − 27·27-s + 214.·29-s + 88.6·31-s + 73.7·33-s + 35·35-s + 162.·37-s + 105.·39-s − 337.·41-s − 122.·43-s − 45·45-s − 354.·47-s + 49·49-s + 55.2·51-s + 676.·53-s + 122.·55-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.447·5-s − 0.377·7-s + 0.333·9-s − 0.674·11-s − 0.747·13-s + 0.258·15-s − 0.262·17-s + 0.813·19-s + 0.218·21-s + 1.32·23-s + 0.200·25-s − 0.192·27-s + 1.37·29-s + 0.513·31-s + 0.389·33-s + 0.169·35-s + 0.720·37-s + 0.431·39-s − 1.28·41-s − 0.433·43-s − 0.149·45-s − 1.09·47-s + 0.142·49-s + 0.151·51-s + 1.75·53-s + 0.301·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + 3T \) |
| 5 | \( 1 + 5T \) |
| 7 | \( 1 + 7T \) |
good | 11 | \( 1 + 24.5T + 1.33e3T^{2} \) |
| 13 | \( 1 + 35.0T + 2.19e3T^{2} \) |
| 17 | \( 1 + 18.4T + 4.91e3T^{2} \) |
| 19 | \( 1 - 67.4T + 6.85e3T^{2} \) |
| 23 | \( 1 - 145.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 214.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 88.6T + 2.97e4T^{2} \) |
| 37 | \( 1 - 162.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 337.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 122.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 354.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 676.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 501.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 708.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 907.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 430.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 41.3T + 3.89e5T^{2} \) |
| 79 | \( 1 + 890.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.05e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.47e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 555.T + 9.12e5T^{2} \) |
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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
−8.556201262653392522003485250867, −7.68966543785456998591891239595, −6.98447691886513795193893908172, −6.25501745590821073112018368398, −5.10018135428180571689261152901, −4.71814790612179496745598998928, −3.40296048652726666401617988475, −2.57651878391400688897358158339, −1.05093785090057980065476714016, 0,
1.05093785090057980065476714016, 2.57651878391400688897358158339, 3.40296048652726666401617988475, 4.71814790612179496745598998928, 5.10018135428180571689261152901, 6.25501745590821073112018368398, 6.98447691886513795193893908172, 7.68966543785456998591891239595, 8.556201262653392522003485250867