| L(s) = 1 | − 5.71·3-s − 19.9·5-s − 4.43·7-s + 5.67·9-s + 65.8·13-s + 114.·15-s − 79.1·17-s − 91.2·19-s + 25.3·21-s + 90.4·23-s + 273.·25-s + 121.·27-s + 157.·29-s − 44.8·31-s + 88.5·35-s + 94.0·37-s − 376.·39-s + 104.·41-s + 251.·43-s − 113.·45-s + 538.·47-s − 323.·49-s + 452.·51-s + 588.·53-s + 521.·57-s − 466.·59-s − 506.·61-s + ⋯ |
| L(s) = 1 | − 1.10·3-s − 1.78·5-s − 0.239·7-s + 0.210·9-s + 1.40·13-s + 1.96·15-s − 1.12·17-s − 1.10·19-s + 0.263·21-s + 0.820·23-s + 2.18·25-s + 0.868·27-s + 1.01·29-s − 0.260·31-s + 0.427·35-s + 0.417·37-s − 1.54·39-s + 0.399·41-s + 0.893·43-s − 0.375·45-s + 1.67·47-s − 0.942·49-s + 1.24·51-s + 1.52·53-s + 1.21·57-s − 1.02·59-s − 1.06·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 968 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 968 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 + 5.71T + 27T^{2} \) |
| 5 | \( 1 + 19.9T + 125T^{2} \) |
| 7 | \( 1 + 4.43T + 343T^{2} \) |
| 13 | \( 1 - 65.8T + 2.19e3T^{2} \) |
| 17 | \( 1 + 79.1T + 4.91e3T^{2} \) |
| 19 | \( 1 + 91.2T + 6.85e3T^{2} \) |
| 23 | \( 1 - 90.4T + 1.21e4T^{2} \) |
| 29 | \( 1 - 157.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 44.8T + 2.97e4T^{2} \) |
| 37 | \( 1 - 94.0T + 5.06e4T^{2} \) |
| 41 | \( 1 - 104.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 251.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 538.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 588.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 466.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 506.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 228.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 117.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 632.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 857.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 261.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 441.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.56e3T + 9.12e5T^{2} \) |
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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
−8.857569476021439325256173429898, −8.528580990033672469317640205406, −7.40417523333415210620250790572, −6.60583757547349246449832881958, −5.88962428079821152347838002052, −4.57890102843369794312224202278, −4.08595461492847627485845493437, −2.90978513582651906088486712706, −0.922856477686892987470739972994, 0,
0.922856477686892987470739972994, 2.90978513582651906088486712706, 4.08595461492847627485845493437, 4.57890102843369794312224202278, 5.88962428079821152347838002052, 6.60583757547349246449832881958, 7.40417523333415210620250790572, 8.528580990033672469317640205406, 8.857569476021439325256173429898
