| L(s) = 1 | + 0.779·3-s + 5·5-s − 7·7-s − 26.3·9-s + 55.7·11-s − 46.6·13-s + 3.89·15-s − 32.8·17-s − 35.7·19-s − 5.45·21-s + 82.7·23-s + 25·25-s − 41.5·27-s + 195.·29-s − 94.9·31-s + 43.4·33-s − 35·35-s + 400.·37-s − 36.3·39-s − 72.1·41-s + 189.·43-s − 131.·45-s − 466.·47-s + 49·49-s − 25.6·51-s + 575.·53-s + 278.·55-s + ⋯ |
| L(s) = 1 | + 0.149·3-s + 0.447·5-s − 0.377·7-s − 0.977·9-s + 1.52·11-s − 0.995·13-s + 0.0670·15-s − 0.469·17-s − 0.431·19-s − 0.0566·21-s + 0.749·23-s + 0.200·25-s − 0.296·27-s + 1.24·29-s − 0.550·31-s + 0.229·33-s − 0.169·35-s + 1.78·37-s − 0.149·39-s − 0.274·41-s + 0.673·43-s − 0.437·45-s − 1.44·47-s + 0.142·49-s − 0.0703·51-s + 1.49·53-s + 0.683·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1120 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.094094975\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.094094975\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - 5T \) |
| 7 | \( 1 + 7T \) |
| good | 3 | \( 1 - 0.779T + 27T^{2} \) |
| 11 | \( 1 - 55.7T + 1.33e3T^{2} \) |
| 13 | \( 1 + 46.6T + 2.19e3T^{2} \) |
| 17 | \( 1 + 32.8T + 4.91e3T^{2} \) |
| 19 | \( 1 + 35.7T + 6.85e3T^{2} \) |
| 23 | \( 1 - 82.7T + 1.21e4T^{2} \) |
| 29 | \( 1 - 195.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 94.9T + 2.97e4T^{2} \) |
| 37 | \( 1 - 400.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 72.1T + 6.89e4T^{2} \) |
| 43 | \( 1 - 189.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 466.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 575.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 155.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 35.1T + 2.26e5T^{2} \) |
| 67 | \( 1 - 407.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 766.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 776.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 872.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 759.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 963.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.06e3T + 9.12e5T^{2} \) |
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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
−9.282212564548140217233484471680, −8.901117667092698325659507233993, −7.85971548878074546440772415192, −6.71947524044346397971370321525, −6.27661225184885056115058222605, −5.18202699942552594435116020741, −4.20181854616023223685932812490, −3.06751444599004621198537617939, −2.16290725137243830139623569556, −0.73912593280595603400899401937,
0.73912593280595603400899401937, 2.16290725137243830139623569556, 3.06751444599004621198537617939, 4.20181854616023223685932812490, 5.18202699942552594435116020741, 6.27661225184885056115058222605, 6.71947524044346397971370321525, 7.85971548878074546440772415192, 8.901117667092698325659507233993, 9.282212564548140217233484471680
