| L(s) = 1 | + 1.08·2-s + 0.870·3-s − 0.829·4-s + 0.942·6-s − 4.02·7-s − 3.06·8-s − 2.24·9-s + 11-s − 0.722·12-s + 3.57·13-s − 4.35·14-s − 1.65·16-s − 2.13·17-s − 2.42·18-s − 19-s − 3.50·21-s + 1.08·22-s + 1.52·23-s − 2.66·24-s + 3.86·26-s − 4.56·27-s + 3.34·28-s + 0.640·29-s − 2.79·31-s + 4.33·32-s + 0.870·33-s − 2.31·34-s + ⋯ |
| L(s) = 1 | + 0.764·2-s + 0.502·3-s − 0.414·4-s + 0.384·6-s − 1.52·7-s − 1.08·8-s − 0.747·9-s + 0.301·11-s − 0.208·12-s + 0.990·13-s − 1.16·14-s − 0.413·16-s − 0.518·17-s − 0.571·18-s − 0.229·19-s − 0.765·21-s + 0.230·22-s + 0.318·23-s − 0.544·24-s + 0.757·26-s − 0.878·27-s + 0.631·28-s + 0.118·29-s − 0.502·31-s + 0.766·32-s + 0.151·33-s − 0.396·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.670780962\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.670780962\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 11 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| good | 2 | \( 1 - 1.08T + 2T^{2} \) |
| 3 | \( 1 - 0.870T + 3T^{2} \) |
| 7 | \( 1 + 4.02T + 7T^{2} \) |
| 13 | \( 1 - 3.57T + 13T^{2} \) |
| 17 | \( 1 + 2.13T + 17T^{2} \) |
| 23 | \( 1 - 1.52T + 23T^{2} \) |
| 29 | \( 1 - 0.640T + 29T^{2} \) |
| 31 | \( 1 + 2.79T + 31T^{2} \) |
| 37 | \( 1 + 6.88T + 37T^{2} \) |
| 41 | \( 1 - 2.11T + 41T^{2} \) |
| 43 | \( 1 - 11.2T + 43T^{2} \) |
| 47 | \( 1 - 5.49T + 47T^{2} \) |
| 53 | \( 1 + 10.3T + 53T^{2} \) |
| 59 | \( 1 - 10.5T + 59T^{2} \) |
| 61 | \( 1 - 7.73T + 61T^{2} \) |
| 67 | \( 1 + 7.97T + 67T^{2} \) |
| 71 | \( 1 - 7.60T + 71T^{2} \) |
| 73 | \( 1 + 3.48T + 73T^{2} \) |
| 79 | \( 1 - 5.50T + 79T^{2} \) |
| 83 | \( 1 + 15.4T + 83T^{2} \) |
| 89 | \( 1 - 17.3T + 89T^{2} \) |
| 97 | \( 1 - 6.28T + 97T^{2} \) |
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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
−8.479549177111224492171746644928, −7.38500785169974840451243592956, −6.47999584853260496329935867279, −6.03725720019389929868542866147, −5.38681215936779468982199882348, −4.29778349620316469033799801108, −3.60366371611535553132607532434, −3.17821997791597485073692478646, −2.29183037960428100046486977644, −0.58083800267786977253819997857,
0.58083800267786977253819997857, 2.29183037960428100046486977644, 3.17821997791597485073692478646, 3.60366371611535553132607532434, 4.29778349620316469033799801108, 5.38681215936779468982199882348, 6.03725720019389929868542866147, 6.47999584853260496329935867279, 7.38500785169974840451243592956, 8.479549177111224492171746644928
