| L(s) = 1 | + 1.56·2-s − 5.56·4-s + 7·7-s − 21.1·8-s − 10.2·11-s − 34.3·13-s + 10.9·14-s + 11.4·16-s + 82.6·17-s + 90.7·19-s − 16·22-s + 12.1·23-s − 53.6·26-s − 38.9·28-s − 105.·29-s − 142.·31-s + 187.·32-s + 128.·34-s − 64.8·37-s + 141.·38-s − 195.·41-s + 319.·43-s + 56.9·44-s + 18.9·46-s + 318.·47-s + 49·49-s + 191.·52-s + ⋯ |
| L(s) = 1 | + 0.552·2-s − 0.695·4-s + 0.377·7-s − 0.935·8-s − 0.280·11-s − 0.732·13-s + 0.208·14-s + 0.178·16-s + 1.17·17-s + 1.09·19-s − 0.155·22-s + 0.110·23-s − 0.404·26-s − 0.262·28-s − 0.673·29-s − 0.823·31-s + 1.03·32-s + 0.650·34-s − 0.288·37-s + 0.604·38-s − 0.743·41-s + 1.13·43-s + 0.195·44-s + 0.0607·46-s + 0.989·47-s + 0.142·49-s + 0.509·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - 7T \) |
| good | 2 | \( 1 - 1.56T + 8T^{2} \) |
| 11 | \( 1 + 10.2T + 1.33e3T^{2} \) |
| 13 | \( 1 + 34.3T + 2.19e3T^{2} \) |
| 17 | \( 1 - 82.6T + 4.91e3T^{2} \) |
| 19 | \( 1 - 90.7T + 6.85e3T^{2} \) |
| 23 | \( 1 - 12.1T + 1.21e4T^{2} \) |
| 29 | \( 1 + 105.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 142.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 64.8T + 5.06e4T^{2} \) |
| 41 | \( 1 + 195.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 319.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 318.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 296.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 284T + 2.05e5T^{2} \) |
| 61 | \( 1 + 494.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 549.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 740.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 556.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 376.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 752.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 945.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 180.T + 9.12e5T^{2} \) |
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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
−8.744023972340342636495888052035, −7.74520809825870212450709662147, −7.23040216516996035312538400026, −5.77569506598388400946932151879, −5.39513479529210235057473385686, −4.52874192477245521867390414255, −3.59559330448520989319147365630, −2.73692533154289844972588913230, −1.27633572297257869773287410234, 0,
1.27633572297257869773287410234, 2.73692533154289844972588913230, 3.59559330448520989319147365630, 4.52874192477245521867390414255, 5.39513479529210235057473385686, 5.77569506598388400946932151879, 7.23040216516996035312538400026, 7.74520809825870212450709662147, 8.744023972340342636495888052035
