| L(s) = 1 | + 3-s − 0.697·5-s + 7-s + 9-s − 5.60·11-s + 2.30·13-s − 0.697·15-s − 0.394·17-s + 0.394·19-s + 21-s + 23-s − 4.51·25-s + 27-s − 5.60·29-s + 3.60·31-s − 5.60·33-s − 0.697·35-s + 5.60·37-s + 2.30·39-s + 3.60·41-s − 4.30·43-s − 0.697·45-s + 4.60·47-s + 49-s − 0.394·51-s + 5.90·53-s + 3.90·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.311·5-s + 0.377·7-s + 0.333·9-s − 1.69·11-s + 0.638·13-s − 0.180·15-s − 0.0956·17-s + 0.0904·19-s + 0.218·21-s + 0.208·23-s − 0.902·25-s + 0.192·27-s − 1.04·29-s + 0.647·31-s − 0.975·33-s − 0.117·35-s + 0.921·37-s + 0.368·39-s + 0.563·41-s − 0.656·43-s − 0.103·45-s + 0.671·47-s + 0.142·49-s − 0.0552·51-s + 0.811·53-s + 0.526·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| good | 5 | \( 1 + 0.697T + 5T^{2} \) |
| 11 | \( 1 + 5.60T + 11T^{2} \) |
| 13 | \( 1 - 2.30T + 13T^{2} \) |
| 17 | \( 1 + 0.394T + 17T^{2} \) |
| 19 | \( 1 - 0.394T + 19T^{2} \) |
| 29 | \( 1 + 5.60T + 29T^{2} \) |
| 31 | \( 1 - 3.60T + 31T^{2} \) |
| 37 | \( 1 - 5.60T + 37T^{2} \) |
| 41 | \( 1 - 3.60T + 41T^{2} \) |
| 43 | \( 1 + 4.30T + 43T^{2} \) |
| 47 | \( 1 - 4.60T + 47T^{2} \) |
| 53 | \( 1 - 5.90T + 53T^{2} \) |
| 59 | \( 1 + 3.90T + 59T^{2} \) |
| 61 | \( 1 - 4.90T + 61T^{2} \) |
| 67 | \( 1 + 13.3T + 67T^{2} \) |
| 71 | \( 1 + 12.9T + 71T^{2} \) |
| 73 | \( 1 - 11T + 73T^{2} \) |
| 79 | \( 1 + 13.4T + 79T^{2} \) |
| 83 | \( 1 + 16.2T + 83T^{2} \) |
| 89 | \( 1 + 15.5T + 89T^{2} \) |
| 97 | \( 1 - 3.78T + 97T^{2} \) |
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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
−7.58819764669577473840294260538, −7.13321089636703650037789535853, −5.96043976586946745464538280991, −5.49400082936663662208072761670, −4.55561494966599252611094393178, −3.95686791408776099646720011141, −2.99140726503978375234808577001, −2.39937741186204937223714454672, −1.36419158721870838710341872886, 0,
1.36419158721870838710341872886, 2.39937741186204937223714454672, 2.99140726503978375234808577001, 3.95686791408776099646720011141, 4.55561494966599252611094393178, 5.49400082936663662208072761670, 5.96043976586946745464538280991, 7.13321089636703650037789535853, 7.58819764669577473840294260538
