| L(s) = 1 | − 3·3-s + 18.3·5-s + 14.9·7-s + 9·9-s − 68.5·11-s − 13·13-s − 54.9·15-s − 119.·17-s − 3.07·19-s − 44.9·21-s − 137.·23-s + 210.·25-s − 27·27-s − 8.59·29-s − 316.·31-s + 205.·33-s + 274.·35-s − 111.·37-s + 39·39-s + 176.·41-s + 213.·43-s + 164.·45-s + 545.·47-s − 118.·49-s + 359.·51-s − 618·53-s − 1.25e3·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.63·5-s + 0.808·7-s + 0.333·9-s − 1.88·11-s − 0.277·13-s − 0.946·15-s − 1.70·17-s − 0.0371·19-s − 0.466·21-s − 1.24·23-s + 1.68·25-s − 0.192·27-s − 0.0550·29-s − 1.83·31-s + 1.08·33-s + 1.32·35-s − 0.496·37-s + 0.160·39-s + 0.672·41-s + 0.757·43-s + 0.546·45-s + 1.69·47-s − 0.346·49-s + 0.987·51-s − 1.60·53-s − 3.08·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 3T \) |
| 13 | \( 1 + 13T \) |
| good | 5 | \( 1 - 18.3T + 125T^{2} \) |
| 7 | \( 1 - 14.9T + 343T^{2} \) |
| 11 | \( 1 + 68.5T + 1.33e3T^{2} \) |
| 17 | \( 1 + 119.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 3.07T + 6.85e3T^{2} \) |
| 23 | \( 1 + 137.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 8.59T + 2.43e4T^{2} \) |
| 31 | \( 1 + 316.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 111.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 176.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 213.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 545.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 618T + 1.48e5T^{2} \) |
| 59 | \( 1 + 132.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 205.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 678.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 327.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 903.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 84.1T + 4.93e5T^{2} \) |
| 83 | \( 1 + 595.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.03e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 587.T + 9.12e5T^{2} \) |
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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
−9.900292650951395609306995073386, −9.051375297588083673816078599975, −7.984729384720210595141081947625, −6.99634063263232602317182331449, −5.84982022907409222738625879866, −5.37216031238740643707368296617, −4.45733429004565555497857591221, −2.46120218361617140562175307120, −1.83913346844273192939494531768, 0,
1.83913346844273192939494531768, 2.46120218361617140562175307120, 4.45733429004565555497857591221, 5.37216031238740643707368296617, 5.84982022907409222738625879866, 6.99634063263232602317182331449, 7.984729384720210595141081947625, 9.051375297588083673816078599975, 9.900292650951395609306995073386
