L(s) = 1 | + 1.73·2-s + 0.999·4-s − 2·7-s − 1.73·8-s + 1.26·11-s − 13-s − 3.46·14-s − 5·16-s + 3.46·17-s + 4.19·19-s + 2.19·22-s + 4.73·23-s − 1.73·26-s − 1.99·28-s + 9.46·29-s − 0.196·31-s − 5.19·32-s + 5.99·34-s + 4·37-s + 7.26·38-s + 3.46·41-s − 10.1·43-s + 1.26·44-s + 8.19·46-s + 6·47-s − 3·49-s − 0.999·52-s + ⋯ |
L(s) = 1 | + 1.22·2-s + 0.499·4-s − 0.755·7-s − 0.612·8-s + 0.382·11-s − 0.277·13-s − 0.925·14-s − 1.25·16-s + 0.840·17-s + 0.962·19-s + 0.468·22-s + 0.986·23-s − 0.339·26-s − 0.377·28-s + 1.75·29-s − 0.0352·31-s − 0.918·32-s + 1.02·34-s + 0.657·37-s + 1.17·38-s + 0.541·41-s − 1.55·43-s + 0.191·44-s + 1.20·46-s + 0.875·47-s − 0.428·49-s − 0.138·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2925 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.046653647\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.046653647\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 + T \) |
good | 2 | \( 1 - 1.73T + 2T^{2} \) |
| 7 | \( 1 + 2T + 7T^{2} \) |
| 11 | \( 1 - 1.26T + 11T^{2} \) |
| 17 | \( 1 - 3.46T + 17T^{2} \) |
| 19 | \( 1 - 4.19T + 19T^{2} \) |
| 23 | \( 1 - 4.73T + 23T^{2} \) |
| 29 | \( 1 - 9.46T + 29T^{2} \) |
| 31 | \( 1 + 0.196T + 31T^{2} \) |
| 37 | \( 1 - 4T + 37T^{2} \) |
| 41 | \( 1 - 3.46T + 41T^{2} \) |
| 43 | \( 1 + 10.1T + 43T^{2} \) |
| 47 | \( 1 - 6T + 47T^{2} \) |
| 53 | \( 1 + 10.3T + 53T^{2} \) |
| 59 | \( 1 - 15.1T + 59T^{2} \) |
| 61 | \( 1 - 12.3T + 61T^{2} \) |
| 67 | \( 1 - 14.3T + 67T^{2} \) |
| 71 | \( 1 + 1.26T + 71T^{2} \) |
| 73 | \( 1 - 4T + 73T^{2} \) |
| 79 | \( 1 - 12.3T + 79T^{2} \) |
| 83 | \( 1 + 6T + 83T^{2} \) |
| 89 | \( 1 + 0.928T + 89T^{2} \) |
| 97 | \( 1 + 2T + 97T^{2} \) |
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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
−8.786037222986387808377309991904, −7.931730881969451018152222332372, −6.81885331020596073084993053375, −6.48969913887328315967936266671, −5.43296876009771202794452619988, −4.99233904699360916033614303686, −3.96305620360375342839915495084, −3.26138368747094679534477527717, −2.59164491420761569118433029921, −0.905904564982040924484856243921,
0.905904564982040924484856243921, 2.59164491420761569118433029921, 3.26138368747094679534477527717, 3.96305620360375342839915495084, 4.99233904699360916033614303686, 5.43296876009771202794452619988, 6.48969913887328315967936266671, 6.81885331020596073084993053375, 7.931730881969451018152222332372, 8.786037222986387808377309991904