| L(s) = 1 | + 2·2-s + 4·4-s + 8.06·5-s − 26.0·7-s + 8·8-s + 16.1·10-s − 3.26·13-s − 52.1·14-s + 16·16-s + 20.8·17-s − 125.·19-s + 32.2·20-s − 97.8·23-s − 60.0·25-s − 6.52·26-s − 104.·28-s + 263.·29-s + 199.·31-s + 32·32-s + 41.7·34-s − 210.·35-s + 365.·37-s − 251.·38-s + 64.4·40-s − 273.·41-s + 388.·43-s − 195.·46-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s + 0.721·5-s − 1.40·7-s + 0.353·8-s + 0.509·10-s − 0.0696·13-s − 0.995·14-s + 0.250·16-s + 0.297·17-s − 1.52·19-s + 0.360·20-s − 0.886·23-s − 0.480·25-s − 0.0492·26-s − 0.704·28-s + 1.68·29-s + 1.15·31-s + 0.176·32-s + 0.210·34-s − 1.01·35-s + 1.62·37-s − 1.07·38-s + 0.254·40-s − 1.04·41-s + 1.37·43-s − 0.627·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2178 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2178 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.182339007\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.182339007\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 2T \) |
| 3 | \( 1 \) |
| 11 | \( 1 \) |
| good | 5 | \( 1 - 8.06T + 125T^{2} \) |
| 7 | \( 1 + 26.0T + 343T^{2} \) |
| 13 | \( 1 + 3.26T + 2.19e3T^{2} \) |
| 17 | \( 1 - 20.8T + 4.91e3T^{2} \) |
| 19 | \( 1 + 125.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 97.8T + 1.21e4T^{2} \) |
| 29 | \( 1 - 263.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 199.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 365.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 273.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 388.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 51.8T + 1.03e5T^{2} \) |
| 53 | \( 1 - 412.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 26.2T + 2.05e5T^{2} \) |
| 61 | \( 1 - 164.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 276.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 516.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 241.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 273.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 72.5T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.19e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.46e3T + 9.12e5T^{2} \) |
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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
−8.724101855411699127697479587803, −7.895029125846949557224902596895, −6.74849405008713167731107136652, −6.27330877748375966315196642359, −5.78642002279017728799383062536, −4.60078882242618590123312764323, −3.86027603887626369542259601273, −2.81004274623799271759502110279, −2.17801370924688720620042714360, −0.69932308678081320831175326263,
0.69932308678081320831175326263, 2.17801370924688720620042714360, 2.81004274623799271759502110279, 3.86027603887626369542259601273, 4.60078882242618590123312764323, 5.78642002279017728799383062536, 6.27330877748375966315196642359, 6.74849405008713167731107136652, 7.895029125846949557224902596895, 8.724101855411699127697479587803
