| L(s) = 1 | + 1.64·2-s + 3-s + 0.711·4-s + 5-s + 1.64·6-s + 4.47·7-s − 2.12·8-s + 9-s + 1.64·10-s − 3.44·11-s + 0.711·12-s − 2.47·13-s + 7.37·14-s + 15-s − 4.91·16-s + 7.62·17-s + 1.64·18-s + 0.711·20-s + 4.47·21-s − 5.66·22-s + 3.87·23-s − 2.12·24-s + 25-s − 4.08·26-s + 27-s + 3.18·28-s + 8.73·29-s + ⋯ |
| L(s) = 1 | + 1.16·2-s + 0.577·3-s + 0.355·4-s + 0.447·5-s + 0.672·6-s + 1.69·7-s − 0.750·8-s + 0.333·9-s + 0.520·10-s − 1.03·11-s + 0.205·12-s − 0.687·13-s + 1.97·14-s + 0.258·15-s − 1.22·16-s + 1.85·17-s + 0.388·18-s + 0.159·20-s + 0.977·21-s − 1.20·22-s + 0.807·23-s − 0.433·24-s + 0.200·25-s − 0.800·26-s + 0.192·27-s + 0.602·28-s + 1.62·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5415 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5415 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.505951076\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.505951076\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 - 1.64T + 2T^{2} \) |
| 7 | \( 1 - 4.47T + 7T^{2} \) |
| 11 | \( 1 + 3.44T + 11T^{2} \) |
| 13 | \( 1 + 2.47T + 13T^{2} \) |
| 17 | \( 1 - 7.62T + 17T^{2} \) |
| 23 | \( 1 - 3.87T + 23T^{2} \) |
| 29 | \( 1 - 8.73T + 29T^{2} \) |
| 31 | \( 1 - 0.422T + 31T^{2} \) |
| 37 | \( 1 + 3.90T + 37T^{2} \) |
| 41 | \( 1 + 5.29T + 41T^{2} \) |
| 43 | \( 1 - 2.47T + 43T^{2} \) |
| 47 | \( 1 + 0.677T + 47T^{2} \) |
| 53 | \( 1 + 11.4T + 53T^{2} \) |
| 59 | \( 1 - 8.53T + 59T^{2} \) |
| 61 | \( 1 - 8.20T + 61T^{2} \) |
| 67 | \( 1 - 9.63T + 67T^{2} \) |
| 71 | \( 1 - 3.85T + 71T^{2} \) |
| 73 | \( 1 - 16.7T + 73T^{2} \) |
| 79 | \( 1 + 12.1T + 79T^{2} \) |
| 83 | \( 1 + 2.03T + 83T^{2} \) |
| 89 | \( 1 - 3.14T + 89T^{2} \) |
| 97 | \( 1 + 2T + 97T^{2} \) |
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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
−8.200662523559121529940470755862, −7.50678870452413745689915333384, −6.68481851290878416373411730448, −5.57707791431417286126288211030, −5.04064581241648262589470066283, −4.83241165188445837318853061388, −3.72095512132286507300597668960, −2.89009359418090848730770209898, −2.23576052236733492407079012232, −1.10305297478285514829812944436,
1.10305297478285514829812944436, 2.23576052236733492407079012232, 2.89009359418090848730770209898, 3.72095512132286507300597668960, 4.83241165188445837318853061388, 5.04064581241648262589470066283, 5.57707791431417286126288211030, 6.68481851290878416373411730448, 7.50678870452413745689915333384, 8.200662523559121529940470755862
