| L(s) = 1 | − 2.28·2-s − 2.13·3-s + 3.22·4-s + 4.88·6-s − 7-s − 2.80·8-s + 1.56·9-s − 3.22·11-s − 6.90·12-s + 1.60·13-s + 2.28·14-s − 0.0344·16-s + 2.87·17-s − 3.58·18-s − 4.80·19-s + 2.13·21-s + 7.37·22-s − 23-s + 6.00·24-s − 3.66·26-s + 3.05·27-s − 3.22·28-s + 1.87·29-s + 3.43·31-s + 5.69·32-s + 6.89·33-s − 6.58·34-s + ⋯ |
| L(s) = 1 | − 1.61·2-s − 1.23·3-s + 1.61·4-s + 1.99·6-s − 0.377·7-s − 0.993·8-s + 0.522·9-s − 0.972·11-s − 1.99·12-s + 0.443·13-s + 0.611·14-s − 0.00860·16-s + 0.698·17-s − 0.845·18-s − 1.10·19-s + 0.466·21-s + 1.57·22-s − 0.208·23-s + 1.22·24-s − 0.717·26-s + 0.588·27-s − 0.610·28-s + 0.347·29-s + 0.617·31-s + 1.00·32-s + 1.20·33-s − 1.12·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| good | 2 | \( 1 + 2.28T + 2T^{2} \) |
| 3 | \( 1 + 2.13T + 3T^{2} \) |
| 11 | \( 1 + 3.22T + 11T^{2} \) |
| 13 | \( 1 - 1.60T + 13T^{2} \) |
| 17 | \( 1 - 2.87T + 17T^{2} \) |
| 19 | \( 1 + 4.80T + 19T^{2} \) |
| 29 | \( 1 - 1.87T + 29T^{2} \) |
| 31 | \( 1 - 3.43T + 31T^{2} \) |
| 37 | \( 1 + 1.21T + 37T^{2} \) |
| 41 | \( 1 - 2.67T + 41T^{2} \) |
| 43 | \( 1 + 4.93T + 43T^{2} \) |
| 47 | \( 1 + 12.6T + 47T^{2} \) |
| 53 | \( 1 - 6.12T + 53T^{2} \) |
| 59 | \( 1 - 5.88T + 59T^{2} \) |
| 61 | \( 1 - 7.75T + 61T^{2} \) |
| 67 | \( 1 - 1.68T + 67T^{2} \) |
| 71 | \( 1 - 5.28T + 71T^{2} \) |
| 73 | \( 1 + 8.29T + 73T^{2} \) |
| 79 | \( 1 + 7.14T + 79T^{2} \) |
| 83 | \( 1 + 9.49T + 83T^{2} \) |
| 89 | \( 1 + 1.12T + 89T^{2} \) |
| 97 | \( 1 + 1.29T + 97T^{2} \) |
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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
−8.342829697458726292973508418877, −7.41161850528037799917086896187, −6.67577248057656061259376281856, −6.12411495152080983778764942406, −5.33659199148554832536969301922, −4.44887407959672789823067114276, −3.11459104400411978021113441639, −2.06123777749959631159359141869, −0.886188923180332178682307887730, 0,
0.886188923180332178682307887730, 2.06123777749959631159359141869, 3.11459104400411978021113441639, 4.44887407959672789823067114276, 5.33659199148554832536969301922, 6.12411495152080983778764942406, 6.67577248057656061259376281856, 7.41161850528037799917086896187, 8.342829697458726292973508418877
