| L(s) = 1 | + 0.0908·2-s − 1.99·4-s − 5-s + 2.45·7-s − 0.362·8-s − 0.0908·10-s − 0.323·13-s + 0.223·14-s + 3.95·16-s + 3.66·17-s − 0.541·19-s + 1.99·20-s + 4.62·23-s + 25-s − 0.0294·26-s − 4.89·28-s + 1.47·29-s + 0.890·31-s + 1.08·32-s + 0.333·34-s − 2.45·35-s − 2.28·37-s − 0.0491·38-s + 0.362·40-s − 8.98·41-s − 6.31·43-s + 0.420·46-s + ⋯ |
| L(s) = 1 | + 0.0642·2-s − 0.995·4-s − 0.447·5-s + 0.928·7-s − 0.128·8-s − 0.0287·10-s − 0.0897·13-s + 0.0596·14-s + 0.987·16-s + 0.888·17-s − 0.124·19-s + 0.445·20-s + 0.964·23-s + 0.200·25-s − 0.00576·26-s − 0.924·28-s + 0.274·29-s + 0.160·31-s + 0.191·32-s + 0.0571·34-s − 0.415·35-s − 0.375·37-s − 0.00797·38-s + 0.0573·40-s − 1.40·41-s − 0.963·43-s + 0.0619·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5445 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5445 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.544932052\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.544932052\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 0.0908T + 2T^{2} \) |
| 7 | \( 1 - 2.45T + 7T^{2} \) |
| 13 | \( 1 + 0.323T + 13T^{2} \) |
| 17 | \( 1 - 3.66T + 17T^{2} \) |
| 19 | \( 1 + 0.541T + 19T^{2} \) |
| 23 | \( 1 - 4.62T + 23T^{2} \) |
| 29 | \( 1 - 1.47T + 29T^{2} \) |
| 31 | \( 1 - 0.890T + 31T^{2} \) |
| 37 | \( 1 + 2.28T + 37T^{2} \) |
| 41 | \( 1 + 8.98T + 41T^{2} \) |
| 43 | \( 1 + 6.31T + 43T^{2} \) |
| 47 | \( 1 - 1.46T + 47T^{2} \) |
| 53 | \( 1 - 10.3T + 53T^{2} \) |
| 59 | \( 1 + 6.29T + 59T^{2} \) |
| 61 | \( 1 - 6.47T + 61T^{2} \) |
| 67 | \( 1 + 7.05T + 67T^{2} \) |
| 71 | \( 1 - 9.31T + 71T^{2} \) |
| 73 | \( 1 + 6.19T + 73T^{2} \) |
| 79 | \( 1 + 11.6T + 79T^{2} \) |
| 83 | \( 1 - 17.3T + 83T^{2} \) |
| 89 | \( 1 + 4.70T + 89T^{2} \) |
| 97 | \( 1 - 11.0T + 97T^{2} \) |
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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
−8.309990301731805492048605364713, −7.58975309883817955699440220554, −6.87714325071493712277044204083, −5.80151451126408744698753376886, −5.04664834197017441959478126245, −4.67217339372940061232873658393, −3.73309260783977270481078946877, −3.06455485011776347008247248921, −1.69607089937638489003293402702, −0.69776910967023247592411533118,
0.69776910967023247592411533118, 1.69607089937638489003293402702, 3.06455485011776347008247248921, 3.73309260783977270481078946877, 4.67217339372940061232873658393, 5.04664834197017441959478126245, 5.80151451126408744698753376886, 6.87714325071493712277044204083, 7.58975309883817955699440220554, 8.309990301731805492048605364713
