| L(s) = 1 | − 2.23·2-s + 3.00·4-s − 1.61·5-s − 7-s − 2.23·8-s + 3.61·10-s + 1.23·13-s + 2.23·14-s − 0.999·16-s − 1.85·17-s + 3.85·19-s − 4.85·20-s − 6.61·23-s − 2.38·25-s − 2.76·26-s − 3.00·28-s − 6·29-s + 8.09·31-s + 6.70·32-s + 4.14·34-s + 1.61·35-s + 2.38·37-s − 8.61·38-s + 3.61·40-s + 9.61·41-s + 3.70·43-s + 14.7·46-s + ⋯ |
| L(s) = 1 | − 1.58·2-s + 1.50·4-s − 0.723·5-s − 0.377·7-s − 0.790·8-s + 1.14·10-s + 0.342·13-s + 0.597·14-s − 0.249·16-s − 0.449·17-s + 0.884·19-s − 1.08·20-s − 1.37·23-s − 0.476·25-s − 0.542·26-s − 0.566·28-s − 1.11·29-s + 1.45·31-s + 1.18·32-s + 0.711·34-s + 0.273·35-s + 0.391·37-s − 1.39·38-s + 0.572·40-s + 1.50·41-s + 0.565·43-s + 2.18·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 2.23T + 2T^{2} \) |
| 5 | \( 1 + 1.61T + 5T^{2} \) |
| 13 | \( 1 - 1.23T + 13T^{2} \) |
| 17 | \( 1 + 1.85T + 17T^{2} \) |
| 19 | \( 1 - 3.85T + 19T^{2} \) |
| 23 | \( 1 + 6.61T + 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 - 8.09T + 31T^{2} \) |
| 37 | \( 1 - 2.38T + 37T^{2} \) |
| 41 | \( 1 - 9.61T + 41T^{2} \) |
| 43 | \( 1 - 3.70T + 43T^{2} \) |
| 47 | \( 1 - 4.47T + 47T^{2} \) |
| 53 | \( 1 + 11.2T + 53T^{2} \) |
| 59 | \( 1 - 1.23T + 59T^{2} \) |
| 61 | \( 1 + 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 + 12.9T + 71T^{2} \) |
| 73 | \( 1 + 0.291T + 73T^{2} \) |
| 79 | \( 1 - 10T + 79T^{2} \) |
| 83 | \( 1 - 7.52T + 83T^{2} \) |
| 89 | \( 1 + 3.38T + 89T^{2} \) |
| 97 | \( 1 + 6T + 97T^{2} \) |
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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
−7.83531624886157451704184118009, −7.17836938150446953855790095010, −6.33818024900615941525570494929, −5.77271062484641258152234003149, −4.50997669689965905565450356294, −3.88600401062638099245993358410, −2.87294819263312577865075779693, −1.99097364414693392333396811086, −0.941300924108741396789702403481, 0,
0.941300924108741396789702403481, 1.99097364414693392333396811086, 2.87294819263312577865075779693, 3.88600401062638099245993358410, 4.50997669689965905565450356294, 5.77271062484641258152234003149, 6.33818024900615941525570494929, 7.17836938150446953855790095010, 7.83531624886157451704184118009
