| L(s) = 1 | + 1.57·2-s + 3.08·3-s + 0.484·4-s − 5-s + 4.86·6-s − 2.38·8-s + 6.52·9-s − 1.57·10-s − 3.00·11-s + 1.49·12-s − 1.53·13-s − 3.08·15-s − 4.73·16-s + 0.0169·17-s + 10.2·18-s − 3.56·19-s − 0.484·20-s − 4.73·22-s − 2.90·23-s − 7.37·24-s + 25-s − 2.41·26-s + 10.8·27-s − 1.69·29-s − 4.86·30-s − 5.42·31-s − 2.68·32-s + ⋯ |
| L(s) = 1 | + 1.11·2-s + 1.78·3-s + 0.242·4-s − 0.447·5-s + 1.98·6-s − 0.844·8-s + 2.17·9-s − 0.498·10-s − 0.905·11-s + 0.431·12-s − 0.425·13-s − 0.796·15-s − 1.18·16-s + 0.00410·17-s + 2.42·18-s − 0.817·19-s − 0.108·20-s − 1.00·22-s − 0.605·23-s − 1.50·24-s + 0.200·25-s − 0.474·26-s + 2.09·27-s − 0.315·29-s − 0.887·30-s − 0.974·31-s − 0.474·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9065 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9065 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| 37 | \( 1 + T \) |
| good | 2 | \( 1 - 1.57T + 2T^{2} \) |
| 3 | \( 1 - 3.08T + 3T^{2} \) |
| 11 | \( 1 + 3.00T + 11T^{2} \) |
| 13 | \( 1 + 1.53T + 13T^{2} \) |
| 17 | \( 1 - 0.0169T + 17T^{2} \) |
| 19 | \( 1 + 3.56T + 19T^{2} \) |
| 23 | \( 1 + 2.90T + 23T^{2} \) |
| 29 | \( 1 + 1.69T + 29T^{2} \) |
| 31 | \( 1 + 5.42T + 31T^{2} \) |
| 41 | \( 1 - 3.01T + 41T^{2} \) |
| 43 | \( 1 + 5.46T + 43T^{2} \) |
| 47 | \( 1 + 5.78T + 47T^{2} \) |
| 53 | \( 1 - 9.55T + 53T^{2} \) |
| 59 | \( 1 - 8.22T + 59T^{2} \) |
| 61 | \( 1 + 9.85T + 61T^{2} \) |
| 67 | \( 1 - 3.92T + 67T^{2} \) |
| 71 | \( 1 + 15.5T + 71T^{2} \) |
| 73 | \( 1 + 4.91T + 73T^{2} \) |
| 79 | \( 1 + 1.28T + 79T^{2} \) |
| 83 | \( 1 + 13.2T + 83T^{2} \) |
| 89 | \( 1 - 2.58T + 89T^{2} \) |
| 97 | \( 1 - 2.85T + 97T^{2} \) |
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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
−7.44954220177561483551045021129, −6.85269762194213492205441756128, −5.86418057494531488524776071013, −5.07153895874306811986517781762, −4.31807650494127170076522469594, −3.86341573891025088815201955777, −3.12297385040956041887717801095, −2.56905690658309939541332498994, −1.81153034165271672154755648750, 0,
1.81153034165271672154755648750, 2.56905690658309939541332498994, 3.12297385040956041887717801095, 3.86341573891025088815201955777, 4.31807650494127170076522469594, 5.07153895874306811986517781762, 5.86418057494531488524776071013, 6.85269762194213492205441756128, 7.44954220177561483551045021129
