| L(s) = 1 | + 0.706·3-s − 5-s − 5.21·7-s − 2.50·9-s + 4.73·11-s + 0.250·13-s − 0.706·15-s − 2.11·17-s + 5.84·19-s − 3.68·21-s − 23-s + 25-s − 3.88·27-s + 6.19·29-s + 3.60·31-s + 3.34·33-s + 5.21·35-s − 10.8·37-s + 0.177·39-s + 1.02·41-s + 11.6·43-s + 2.50·45-s − 0.134·47-s + 20.1·49-s − 1.49·51-s + 9.78·53-s − 4.73·55-s + ⋯ |
| L(s) = 1 | + 0.408·3-s − 0.447·5-s − 1.97·7-s − 0.833·9-s + 1.42·11-s + 0.0696·13-s − 0.182·15-s − 0.512·17-s + 1.34·19-s − 0.803·21-s − 0.208·23-s + 0.200·25-s − 0.748·27-s + 1.14·29-s + 0.648·31-s + 0.582·33-s + 0.881·35-s − 1.78·37-s + 0.0284·39-s + 0.159·41-s + 1.77·43-s + 0.372·45-s − 0.0196·47-s + 2.88·49-s − 0.209·51-s + 1.34·53-s − 0.638·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| good | 3 | \( 1 - 0.706T + 3T^{2} \) |
| 7 | \( 1 + 5.21T + 7T^{2} \) |
| 11 | \( 1 - 4.73T + 11T^{2} \) |
| 13 | \( 1 - 0.250T + 13T^{2} \) |
| 17 | \( 1 + 2.11T + 17T^{2} \) |
| 19 | \( 1 - 5.84T + 19T^{2} \) |
| 29 | \( 1 - 6.19T + 29T^{2} \) |
| 31 | \( 1 - 3.60T + 31T^{2} \) |
| 37 | \( 1 + 10.8T + 37T^{2} \) |
| 41 | \( 1 - 1.02T + 41T^{2} \) |
| 43 | \( 1 - 11.6T + 43T^{2} \) |
| 47 | \( 1 + 0.134T + 47T^{2} \) |
| 53 | \( 1 - 9.78T + 53T^{2} \) |
| 59 | \( 1 + 10.3T + 59T^{2} \) |
| 61 | \( 1 + 1.05T + 61T^{2} \) |
| 67 | \( 1 + 11.2T + 67T^{2} \) |
| 71 | \( 1 + 10.4T + 71T^{2} \) |
| 73 | \( 1 + 6.98T + 73T^{2} \) |
| 79 | \( 1 - 6.07T + 79T^{2} \) |
| 83 | \( 1 + 17.6T + 83T^{2} \) |
| 89 | \( 1 - 0.300T + 89T^{2} \) |
| 97 | \( 1 - 0.307T + 97T^{2} \) |
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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
−7.34012337177708709766478963655, −6.94870543560697958220498147322, −6.14096002096924923883856894260, −5.75695864828326471346046780575, −4.48820923138682718976706978380, −3.73256099476610854908726656479, −3.17033281947353231456299148302, −2.60974333100114836473340189767, −1.12771983226195061090957751720, 0,
1.12771983226195061090957751720, 2.60974333100114836473340189767, 3.17033281947353231456299148302, 3.73256099476610854908726656479, 4.48820923138682718976706978380, 5.75695864828326471346046780575, 6.14096002096924923883856894260, 6.94870543560697958220498147322, 7.34012337177708709766478963655
