| L(s) = 1 | − 3-s − 2.87·5-s + 4.47·7-s + 9-s − 4.99·11-s − 13-s + 2.87·15-s − 1.39·17-s + 5.86·19-s − 4.47·21-s + 1.15·23-s + 3.24·25-s − 27-s + 2.24·29-s − 1.27·31-s + 4.99·33-s − 12.8·35-s − 11.1·37-s + 39-s − 0.327·41-s + 5.63·43-s − 2.87·45-s − 3.59·47-s + 12.9·49-s + 1.39·51-s + 9.63·53-s + 14.3·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.28·5-s + 1.68·7-s + 0.333·9-s − 1.50·11-s − 0.277·13-s + 0.741·15-s − 0.337·17-s + 1.34·19-s − 0.975·21-s + 0.239·23-s + 0.648·25-s − 0.192·27-s + 0.416·29-s − 0.228·31-s + 0.869·33-s − 2.16·35-s − 1.83·37-s + 0.160·39-s − 0.0511·41-s + 0.859·43-s − 0.427·45-s − 0.525·47-s + 1.85·49-s + 0.195·51-s + 1.32·53-s + 1.93·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9984 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9984 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| good | 5 | \( 1 + 2.87T + 5T^{2} \) |
| 7 | \( 1 - 4.47T + 7T^{2} \) |
| 11 | \( 1 + 4.99T + 11T^{2} \) |
| 17 | \( 1 + 1.39T + 17T^{2} \) |
| 19 | \( 1 - 5.86T + 19T^{2} \) |
| 23 | \( 1 - 1.15T + 23T^{2} \) |
| 29 | \( 1 - 2.24T + 29T^{2} \) |
| 31 | \( 1 + 1.27T + 31T^{2} \) |
| 37 | \( 1 + 11.1T + 37T^{2} \) |
| 41 | \( 1 + 0.327T + 41T^{2} \) |
| 43 | \( 1 - 5.63T + 43T^{2} \) |
| 47 | \( 1 + 3.59T + 47T^{2} \) |
| 53 | \( 1 - 9.63T + 53T^{2} \) |
| 59 | \( 1 + 0.749T + 59T^{2} \) |
| 61 | \( 1 + 4.24T + 61T^{2} \) |
| 67 | \( 1 - 9.30T + 67T^{2} \) |
| 71 | \( 1 - 11.4T + 71T^{2} \) |
| 73 | \( 1 + 15.1T + 73T^{2} \) |
| 79 | \( 1 + 7.19T + 79T^{2} \) |
| 83 | \( 1 + 8.74T + 83T^{2} \) |
| 89 | \( 1 - 5.02T + 89T^{2} \) |
| 97 | \( 1 - 17.4T + 97T^{2} \) |
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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
−7.41728155665073313001538417796, −6.95356353517030342028985993527, −5.61950336141195256925003406599, −5.18604843595957692193912463804, −4.70637192382584517306512418245, −3.97760010748736495650339995909, −3.05800207637193644802155843169, −2.09869349127938317062484562532, −1.05356038914520338707285940144, 0,
1.05356038914520338707285940144, 2.09869349127938317062484562532, 3.05800207637193644802155843169, 3.97760010748736495650339995909, 4.70637192382584517306512418245, 5.18604843595957692193912463804, 5.61950336141195256925003406599, 6.95356353517030342028985993527, 7.41728155665073313001538417796
