L(s) = 1 | + 2-s + 2.94·3-s + 4-s + 2.36·5-s + 2.94·6-s − 1.66·7-s + 8-s + 5.68·9-s + 2.36·10-s + 2.94·12-s − 0.633·13-s − 1.66·14-s + 6.97·15-s + 16-s + 0.581·17-s + 5.68·18-s − 19-s + 2.36·20-s − 4.89·21-s + 3.60·23-s + 2.94·24-s + 0.599·25-s − 0.633·26-s + 7.92·27-s − 1.66·28-s + 9.13·29-s + 6.97·30-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1.70·3-s + 0.5·4-s + 1.05·5-s + 1.20·6-s − 0.627·7-s + 0.353·8-s + 1.89·9-s + 0.748·10-s + 0.850·12-s − 0.175·13-s − 0.443·14-s + 1.80·15-s + 0.250·16-s + 0.140·17-s + 1.34·18-s − 0.229·19-s + 0.529·20-s − 1.06·21-s + 0.752·23-s + 0.601·24-s + 0.119·25-s − 0.124·26-s + 1.52·27-s − 0.313·28-s + 1.69·29-s + 1.27·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4598 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4598 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(6.854980223\) |
\(L(\frac12)\) |
\(\approx\) |
\(6.854980223\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 11 | \( 1 \) |
| 19 | \( 1 + T \) |
good | 3 | \( 1 - 2.94T + 3T^{2} \) |
| 5 | \( 1 - 2.36T + 5T^{2} \) |
| 7 | \( 1 + 1.66T + 7T^{2} \) |
| 13 | \( 1 + 0.633T + 13T^{2} \) |
| 17 | \( 1 - 0.581T + 17T^{2} \) |
| 23 | \( 1 - 3.60T + 23T^{2} \) |
| 29 | \( 1 - 9.13T + 29T^{2} \) |
| 31 | \( 1 - 1.02T + 31T^{2} \) |
| 37 | \( 1 - 2.15T + 37T^{2} \) |
| 41 | \( 1 + 4.40T + 41T^{2} \) |
| 43 | \( 1 + 9.58T + 43T^{2} \) |
| 47 | \( 1 - 8.73T + 47T^{2} \) |
| 53 | \( 1 + 12.5T + 53T^{2} \) |
| 59 | \( 1 - 6.47T + 59T^{2} \) |
| 61 | \( 1 + 1.32T + 61T^{2} \) |
| 67 | \( 1 + 0.393T + 67T^{2} \) |
| 71 | \( 1 + 0.312T + 71T^{2} \) |
| 73 | \( 1 - 9.31T + 73T^{2} \) |
| 79 | \( 1 + 11.2T + 79T^{2} \) |
| 83 | \( 1 - 6.90T + 83T^{2} \) |
| 89 | \( 1 - 0.519T + 89T^{2} \) |
| 97 | \( 1 + 8.37T + 97T^{2} \) |
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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
−8.385498178523726040298081557313, −7.62884887789577333872115687986, −6.73874932293420936920670877945, −6.32353614540242510613657342847, −5.24611987409778721049055125504, −4.48612253371877496917815690897, −3.51452780774816815635101506995, −2.91163334163347272465203448242, −2.29284464026367898102231178250, −1.38658662756059912951491571656,
1.38658662756059912951491571656, 2.29284464026367898102231178250, 2.91163334163347272465203448242, 3.51452780774816815635101506995, 4.48612253371877496917815690897, 5.24611987409778721049055125504, 6.32353614540242510613657342847, 6.73874932293420936920670877945, 7.62884887789577333872115687986, 8.385498178523726040298081557313