| L(s) = 1 | + 3-s − 1.61·5-s − 7-s + 9-s − 3·11-s + 0.618·13-s − 1.61·15-s + 6.70·17-s − 4.23·19-s − 21-s − 23-s − 2.38·25-s + 27-s − 1.76·29-s − 3·31-s − 3·33-s + 1.61·35-s + 9.47·37-s + 0.618·39-s − 11.9·41-s − 9.09·43-s − 1.61·45-s − 9.23·47-s + 49-s + 6.70·51-s − 9.32·53-s + 4.85·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.723·5-s − 0.377·7-s + 0.333·9-s − 0.904·11-s + 0.171·13-s − 0.417·15-s + 1.62·17-s − 0.971·19-s − 0.218·21-s − 0.208·23-s − 0.476·25-s + 0.192·27-s − 0.327·29-s − 0.538·31-s − 0.522·33-s + 0.273·35-s + 1.55·37-s + 0.0989·39-s − 1.86·41-s − 1.38·43-s − 0.241·45-s − 1.34·47-s + 0.142·49-s + 0.939·51-s − 1.28·53-s + 0.654·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1932 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1932 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + 1.61T + 5T^{2} \) |
| 11 | \( 1 + 3T + 11T^{2} \) |
| 13 | \( 1 - 0.618T + 13T^{2} \) |
| 17 | \( 1 - 6.70T + 17T^{2} \) |
| 19 | \( 1 + 4.23T + 19T^{2} \) |
| 29 | \( 1 + 1.76T + 29T^{2} \) |
| 31 | \( 1 + 3T + 31T^{2} \) |
| 37 | \( 1 - 9.47T + 37T^{2} \) |
| 41 | \( 1 + 11.9T + 41T^{2} \) |
| 43 | \( 1 + 9.09T + 43T^{2} \) |
| 47 | \( 1 + 9.23T + 47T^{2} \) |
| 53 | \( 1 + 9.32T + 53T^{2} \) |
| 59 | \( 1 + 9.61T + 59T^{2} \) |
| 61 | \( 1 - 5.85T + 61T^{2} \) |
| 67 | \( 1 - 6.56T + 67T^{2} \) |
| 71 | \( 1 + 4.61T + 71T^{2} \) |
| 73 | \( 1 - 1.76T + 73T^{2} \) |
| 79 | \( 1 + 6.70T + 79T^{2} \) |
| 83 | \( 1 + 7.47T + 83T^{2} \) |
| 89 | \( 1 + 16.7T + 89T^{2} \) |
| 97 | \( 1 + 12.4T + 97T^{2} \) |
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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
−8.558956470719536268682309433030, −8.049486843936738470746652085513, −7.48670368017283528488447982189, −6.50827852513291290224241119764, −5.56887439675100329374204240796, −4.60536603581385114101347172527, −3.60005369286613938794179311800, −3.01344737718619690504388969003, −1.71866424722829054788062165324, 0,
1.71866424722829054788062165324, 3.01344737718619690504388969003, 3.60005369286613938794179311800, 4.60536603581385114101347172527, 5.56887439675100329374204240796, 6.50827852513291290224241119764, 7.48670368017283528488447982189, 8.049486843936738470746652085513, 8.558956470719536268682309433030
