L(s) = 1 | − 1.34·2-s − 0.192·4-s + 7-s + 2.94·8-s + 5.63·11-s + 4.38·13-s − 1.34·14-s − 3.57·16-s − 2.68·17-s + 8.38·19-s − 7.57·22-s − 5.63·23-s − 5.89·26-s − 0.192·28-s − 8.32·29-s + 6·31-s − 1.08·32-s + 3.61·34-s − 3·37-s − 11.2·38-s + 5.37·41-s − 1.38·43-s − 1.08·44-s + 7.57·46-s − 8.58·47-s + 49-s − 0.844·52-s + ⋯ |
L(s) = 1 | − 0.950·2-s − 0.0962·4-s + 0.377·7-s + 1.04·8-s + 1.69·11-s + 1.21·13-s − 0.359·14-s − 0.894·16-s − 0.652·17-s + 1.92·19-s − 1.61·22-s − 1.17·23-s − 1.15·26-s − 0.0363·28-s − 1.54·29-s + 1.07·31-s − 0.191·32-s + 0.619·34-s − 0.493·37-s − 1.82·38-s + 0.839·41-s − 0.211·43-s − 0.163·44-s + 1.11·46-s − 1.25·47-s + 0.142·49-s − 0.117·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.164490126\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.164490126\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
good | 2 | \( 1 + 1.34T + 2T^{2} \) |
| 11 | \( 1 - 5.63T + 11T^{2} \) |
| 13 | \( 1 - 4.38T + 13T^{2} \) |
| 17 | \( 1 + 2.68T + 17T^{2} \) |
| 19 | \( 1 - 8.38T + 19T^{2} \) |
| 23 | \( 1 + 5.63T + 23T^{2} \) |
| 29 | \( 1 + 8.32T + 29T^{2} \) |
| 31 | \( 1 - 6T + 31T^{2} \) |
| 37 | \( 1 + 3T + 37T^{2} \) |
| 41 | \( 1 - 5.37T + 41T^{2} \) |
| 43 | \( 1 + 1.38T + 43T^{2} \) |
| 47 | \( 1 + 8.58T + 47T^{2} \) |
| 53 | \( 1 - 5.37T + 53T^{2} \) |
| 59 | \( 1 - 8.58T + 59T^{2} \) |
| 61 | \( 1 + 61T^{2} \) |
| 67 | \( 1 + 1.38T + 67T^{2} \) |
| 71 | \( 1 + 0.258T + 71T^{2} \) |
| 73 | \( 1 - 6.38T + 73T^{2} \) |
| 79 | \( 1 - 5.38T + 79T^{2} \) |
| 83 | \( 1 - 16.6T + 83T^{2} \) |
| 89 | \( 1 + 13.9T + 89T^{2} \) |
| 97 | \( 1 + 10.7T + 97T^{2} \) |
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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
−9.428326214929496278906557570420, −8.695276300282267980235064811445, −8.062152880616608392028805309002, −7.20001347338017941842448574288, −6.35906924458646378950049246833, −5.37186956800061872403254250516, −4.21688868407250289415225702278, −3.59544194431005454946603803392, −1.79873980277040304719567661905, −0.969875864494723924114778427341,
0.969875864494723924114778427341, 1.79873980277040304719567661905, 3.59544194431005454946603803392, 4.21688868407250289415225702278, 5.37186956800061872403254250516, 6.35906924458646378950049246833, 7.20001347338017941842448574288, 8.062152880616608392028805309002, 8.695276300282267980235064811445, 9.428326214929496278906557570420