| L(s) = 1 | + 2.52·2-s + 3-s + 4.36·4-s − 5-s + 2.52·6-s − 4.67·7-s + 5.96·8-s + 9-s − 2.52·10-s + 1.88·11-s + 4.36·12-s − 4.74·13-s − 11.8·14-s − 15-s + 6.32·16-s − 6.62·17-s + 2.52·18-s − 2.45·19-s − 4.36·20-s − 4.67·21-s + 4.74·22-s + 8.17·23-s + 5.96·24-s + 25-s − 11.9·26-s + 27-s − 20.4·28-s + ⋯ |
| L(s) = 1 | + 1.78·2-s + 0.577·3-s + 2.18·4-s − 0.447·5-s + 1.03·6-s − 1.76·7-s + 2.11·8-s + 0.333·9-s − 0.797·10-s + 0.567·11-s + 1.26·12-s − 1.31·13-s − 3.15·14-s − 0.258·15-s + 1.58·16-s − 1.60·17-s + 0.594·18-s − 0.563·19-s − 0.976·20-s − 1.02·21-s + 1.01·22-s + 1.70·23-s + 1.21·24-s + 0.200·25-s − 2.34·26-s + 0.192·27-s − 3.86·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 401 | \( 1 - T \) |
| good | 2 | \( 1 - 2.52T + 2T^{2} \) |
| 7 | \( 1 + 4.67T + 7T^{2} \) |
| 11 | \( 1 - 1.88T + 11T^{2} \) |
| 13 | \( 1 + 4.74T + 13T^{2} \) |
| 17 | \( 1 + 6.62T + 17T^{2} \) |
| 19 | \( 1 + 2.45T + 19T^{2} \) |
| 23 | \( 1 - 8.17T + 23T^{2} \) |
| 29 | \( 1 + 1.91T + 29T^{2} \) |
| 31 | \( 1 + 4.15T + 31T^{2} \) |
| 37 | \( 1 + 0.979T + 37T^{2} \) |
| 41 | \( 1 + 0.193T + 41T^{2} \) |
| 43 | \( 1 + 10.4T + 43T^{2} \) |
| 47 | \( 1 + 8.55T + 47T^{2} \) |
| 53 | \( 1 + 3.78T + 53T^{2} \) |
| 59 | \( 1 + 11.3T + 59T^{2} \) |
| 61 | \( 1 - 11.3T + 61T^{2} \) |
| 67 | \( 1 - 1.50T + 67T^{2} \) |
| 71 | \( 1 - 5.98T + 71T^{2} \) |
| 73 | \( 1 - 6.24T + 73T^{2} \) |
| 79 | \( 1 - 4.73T + 79T^{2} \) |
| 83 | \( 1 - 11.6T + 83T^{2} \) |
| 89 | \( 1 + 16.9T + 89T^{2} \) |
| 97 | \( 1 + 7.38T + 97T^{2} \) |
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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
−7.18385413135375501375246420954, −6.71928593440931649455318099886, −6.56110012933818798771241267436, −5.36437727216128554137046900516, −4.68659804915916049034888779880, −3.98598869023201927474928034044, −3.30492632285300541467670403675, −2.79554326293905406628237677104, −1.96780020865650685160091836545, 0,
1.96780020865650685160091836545, 2.79554326293905406628237677104, 3.30492632285300541467670403675, 3.98598869023201927474928034044, 4.68659804915916049034888779880, 5.36437727216128554137046900516, 6.56110012933818798771241267436, 6.71928593440931649455318099886, 7.18385413135375501375246420954
