| L(s) = 1 | − 2.79·2-s + 5.79·4-s − 7-s − 10.5·8-s − 5·11-s − 4.58·13-s + 2.79·14-s + 17.9·16-s − 3·17-s − 5.58·19-s + 13.9·22-s − 4·23-s + 12.7·26-s − 5.79·28-s − 29-s + 4·31-s − 28.9·32-s + 8.37·34-s + 4·37-s + 15.5·38-s − 9.16·41-s + 0.417·43-s − 28.9·44-s + 11.1·46-s + 1.41·47-s − 6·49-s − 26.5·52-s + ⋯ |
| L(s) = 1 | − 1.97·2-s + 2.89·4-s − 0.377·7-s − 3.74·8-s − 1.50·11-s − 1.27·13-s + 0.746·14-s + 4.48·16-s − 0.727·17-s − 1.28·19-s + 2.97·22-s − 0.834·23-s + 2.50·26-s − 1.09·28-s − 0.185·29-s + 0.718·31-s − 5.11·32-s + 1.43·34-s + 0.657·37-s + 2.52·38-s − 1.43·41-s + 0.0636·43-s − 4.36·44-s + 1.64·46-s + 0.206·47-s − 0.857·49-s − 3.68·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.05517660816\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.05517660816\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 29 | \( 1 + T \) |
| good | 2 | \( 1 + 2.79T + 2T^{2} \) |
| 7 | \( 1 + T + 7T^{2} \) |
| 11 | \( 1 + 5T + 11T^{2} \) |
| 13 | \( 1 + 4.58T + 13T^{2} \) |
| 17 | \( 1 + 3T + 17T^{2} \) |
| 19 | \( 1 + 5.58T + 19T^{2} \) |
| 23 | \( 1 + 4T + 23T^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 37 | \( 1 - 4T + 37T^{2} \) |
| 41 | \( 1 + 9.16T + 41T^{2} \) |
| 43 | \( 1 - 0.417T + 43T^{2} \) |
| 47 | \( 1 - 1.41T + 47T^{2} \) |
| 53 | \( 1 - 9.58T + 53T^{2} \) |
| 59 | \( 1 + 1.58T + 59T^{2} \) |
| 61 | \( 1 + 14.7T + 61T^{2} \) |
| 67 | \( 1 + 14.1T + 67T^{2} \) |
| 71 | \( 1 - 0.417T + 71T^{2} \) |
| 73 | \( 1 + 4T + 73T^{2} \) |
| 79 | \( 1 + 1.58T + 79T^{2} \) |
| 83 | \( 1 + 2.41T + 83T^{2} \) |
| 89 | \( 1 + 10.5T + 89T^{2} \) |
| 97 | \( 1 + 2.41T + 97T^{2} \) |
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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
−8.047282066625083926153677363600, −7.58323996682719773187939106612, −6.86408261747212613992598763920, −6.26283183258702254996797159448, −5.47598455669551626507552591187, −4.40296851245701943544853258747, −2.96244421677588145304507343437, −2.49142460523023817611201473839, −1.73365530696011626871105468187, −0.15165067973566073751090693890,
0.15165067973566073751090693890, 1.73365530696011626871105468187, 2.49142460523023817611201473839, 2.96244421677588145304507343437, 4.40296851245701943544853258747, 5.47598455669551626507552591187, 6.26283183258702254996797159448, 6.86408261747212613992598763920, 7.58323996682719773187939106612, 8.047282066625083926153677363600
