L(s) = 1 | + 2.60·2-s − 2.79·3-s + 4.79·4-s − 7.27·6-s − 2.60·7-s + 7.27·8-s + 4.79·9-s − 13.3·12-s + 2.06·13-s − 6.79·14-s + 9.37·16-s − 4.66·17-s + 12.4·18-s − 7.27·19-s + 7.27·21-s − 2.20·23-s − 20.3·24-s + 5.37·26-s − 4.99·27-s − 12.4·28-s + 5.75·29-s − 5·31-s + 9.88·32-s − 12.1·34-s + 22.9·36-s + 0.582·37-s − 18.9·38-s + ⋯ |
L(s) = 1 | + 1.84·2-s − 1.61·3-s + 2.39·4-s − 2.96·6-s − 0.984·7-s + 2.57·8-s + 1.59·9-s − 3.86·12-s + 0.571·13-s − 1.81·14-s + 2.34·16-s − 1.13·17-s + 2.94·18-s − 1.66·19-s + 1.58·21-s − 0.460·23-s − 4.14·24-s + 1.05·26-s − 0.962·27-s − 2.35·28-s + 1.06·29-s − 0.898·31-s + 1.74·32-s − 2.08·34-s + 3.82·36-s + 0.0957·37-s − 3.07·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 \) |
good | 2 | \( 1 - 2.60T + 2T^{2} \) |
| 3 | \( 1 + 2.79T + 3T^{2} \) |
| 7 | \( 1 + 2.60T + 7T^{2} \) |
| 13 | \( 1 - 2.06T + 13T^{2} \) |
| 17 | \( 1 + 4.66T + 17T^{2} \) |
| 19 | \( 1 + 7.27T + 19T^{2} \) |
| 23 | \( 1 + 2.20T + 23T^{2} \) |
| 29 | \( 1 - 5.75T + 29T^{2} \) |
| 31 | \( 1 + 5T + 31T^{2} \) |
| 37 | \( 1 - 0.582T + 37T^{2} \) |
| 41 | \( 1 - 7.27T + 41T^{2} \) |
| 43 | \( 1 + 5.21T + 43T^{2} \) |
| 47 | \( 1 + 10.5T + 47T^{2} \) |
| 53 | \( 1 + 2.20T + 53T^{2} \) |
| 59 | \( 1 + 12.1T + 59T^{2} \) |
| 61 | \( 1 - 1.51T + 61T^{2} \) |
| 67 | \( 1 + 5.58T + 67T^{2} \) |
| 71 | \( 1 - 9.16T + 71T^{2} \) |
| 73 | \( 1 + 6.73T + 73T^{2} \) |
| 79 | \( 1 - 8.79T + 79T^{2} \) |
| 83 | \( 1 + 0.543T + 83T^{2} \) |
| 89 | \( 1 - 12.7T + 89T^{2} \) |
| 97 | \( 1 + 13.3T + 97T^{2} \) |
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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
−8.022119788408962787194667565295, −6.71506542037930826598458549234, −6.55221673002952835982412212628, −6.08781523983833724030976276601, −5.26032994689890223889942001943, −4.48804146063059028340765631951, −3.97300480364838044049344998481, −2.89593804408934521839801885803, −1.75629968365430245147695776899, 0,
1.75629968365430245147695776899, 2.89593804408934521839801885803, 3.97300480364838044049344998481, 4.48804146063059028340765631951, 5.26032994689890223889942001943, 6.08781523983833724030976276601, 6.55221673002952835982412212628, 6.71506542037930826598458549234, 8.022119788408962787194667565295