L(s) = 1 | + 2.27·2-s − 2.82·4-s − 7·7-s − 24.6·8-s + 40.7·11-s − 53.2·13-s − 15.9·14-s − 33.4·16-s + 4.54·17-s + 122.·19-s + 92.7·22-s + 131.·23-s − 121.·26-s + 19.7·28-s + 216.·29-s − 251.·31-s + 120.·32-s + 10.3·34-s − 11.8·37-s + 278.·38-s + 111.·41-s − 369.·43-s − 115.·44-s + 298.·46-s − 262.·47-s + 49·49-s + 150.·52-s + ⋯ |
L(s) = 1 | + 0.804·2-s − 0.353·4-s − 0.377·7-s − 1.08·8-s + 1.11·11-s − 1.13·13-s − 0.303·14-s − 0.522·16-s + 0.0649·17-s + 1.48·19-s + 0.898·22-s + 1.19·23-s − 0.914·26-s + 0.133·28-s + 1.38·29-s − 1.45·31-s + 0.668·32-s + 0.0522·34-s − 0.0528·37-s + 1.19·38-s + 0.425·41-s − 1.30·43-s − 0.394·44-s + 0.957·46-s − 0.815·47-s + 0.142·49-s + 0.401·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + 7T \) |
good | 2 | \( 1 - 2.27T + 8T^{2} \) |
| 11 | \( 1 - 40.7T + 1.33e3T^{2} \) |
| 13 | \( 1 + 53.2T + 2.19e3T^{2} \) |
| 17 | \( 1 - 4.54T + 4.91e3T^{2} \) |
| 19 | \( 1 - 122.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 131.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 216.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 251.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 11.8T + 5.06e4T^{2} \) |
| 41 | \( 1 - 111.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 369.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 262.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 567.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 839.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 485.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 333.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 590.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 490.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 121.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 609.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 719.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 637.T + 9.12e5T^{2} \) |
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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
−8.932068889520827437621225974750, −7.73345263277985193941617836602, −6.88794073984370008804533430532, −6.13139433052273228152951024027, −5.11529370742195282845364745914, −4.62644258946915810473079426420, −3.46735538989211370017006397673, −2.92833208023856057722631891038, −1.32205003009627556106295966332, 0,
1.32205003009627556106295966332, 2.92833208023856057722631891038, 3.46735538989211370017006397673, 4.62644258946915810473079426420, 5.11529370742195282845364745914, 6.13139433052273228152951024027, 6.88794073984370008804533430532, 7.73345263277985193941617836602, 8.932068889520827437621225974750