L(s) = 1 | + 0.940·2-s + 0.854·3-s − 1.11·4-s + 5-s + 0.804·6-s + 7-s − 2.93·8-s − 2.26·9-s + 0.940·10-s − 0.953·12-s − 5.89·13-s + 0.940·14-s + 0.854·15-s − 0.525·16-s + 6.46·17-s − 2.13·18-s + 3.18·19-s − 1.11·20-s + 0.854·21-s + 0.468·23-s − 2.50·24-s + 25-s − 5.54·26-s − 4.50·27-s − 1.11·28-s − 0.0101·29-s + 0.804·30-s + ⋯ |
L(s) = 1 | + 0.665·2-s + 0.493·3-s − 0.557·4-s + 0.447·5-s + 0.328·6-s + 0.377·7-s − 1.03·8-s − 0.756·9-s + 0.297·10-s − 0.275·12-s − 1.63·13-s + 0.251·14-s + 0.220·15-s − 0.131·16-s + 1.56·17-s − 0.503·18-s + 0.729·19-s − 0.249·20-s + 0.186·21-s + 0.0977·23-s − 0.511·24-s + 0.200·25-s − 1.08·26-s − 0.866·27-s − 0.210·28-s − 0.00187·29-s + 0.146·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.584465815\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.584465815\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 - 0.940T + 2T^{2} \) |
| 3 | \( 1 - 0.854T + 3T^{2} \) |
| 13 | \( 1 + 5.89T + 13T^{2} \) |
| 17 | \( 1 - 6.46T + 17T^{2} \) |
| 19 | \( 1 - 3.18T + 19T^{2} \) |
| 23 | \( 1 - 0.468T + 23T^{2} \) |
| 29 | \( 1 + 0.0101T + 29T^{2} \) |
| 31 | \( 1 - 3.76T + 31T^{2} \) |
| 37 | \( 1 - 9.86T + 37T^{2} \) |
| 41 | \( 1 - 9.10T + 41T^{2} \) |
| 43 | \( 1 + 12.2T + 43T^{2} \) |
| 47 | \( 1 + 8.76T + 47T^{2} \) |
| 53 | \( 1 - 9.30T + 53T^{2} \) |
| 59 | \( 1 - 10.4T + 59T^{2} \) |
| 61 | \( 1 - 3.75T + 61T^{2} \) |
| 67 | \( 1 - 6.56T + 67T^{2} \) |
| 71 | \( 1 - 6.07T + 71T^{2} \) |
| 73 | \( 1 + 1.27T + 73T^{2} \) |
| 79 | \( 1 - 11.4T + 79T^{2} \) |
| 83 | \( 1 - 12.1T + 83T^{2} \) |
| 89 | \( 1 + 14.8T + 89T^{2} \) |
| 97 | \( 1 - 7.71T + 97T^{2} \) |
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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
−8.182305194353586589135025410896, −7.936739578460875489064498749910, −6.90849237158207906741935737017, −5.88533968336918059285226373203, −5.29216923242456313701282579375, −4.82962641124930390227967181389, −3.76297451052891659653032774393, −2.97398355872775235424762444975, −2.33524234241780629532574998949, −0.800463899913976810419887841736,
0.800463899913976810419887841736, 2.33524234241780629532574998949, 2.97398355872775235424762444975, 3.76297451052891659653032774393, 4.82962641124930390227967181389, 5.29216923242456313701282579375, 5.88533968336918059285226373203, 6.90849237158207906741935737017, 7.936739578460875489064498749910, 8.182305194353586589135025410896