| L(s) = 1 | − 1.77·2-s + 1.15·4-s − 2.51·7-s + 1.50·8-s − 3.40·11-s − 1.11·13-s + 4.45·14-s − 4.97·16-s + 2.56·17-s + 6.79·19-s + 6.03·22-s − 4.06·23-s + 1.97·26-s − 2.89·28-s + 1.89·29-s − 31-s + 5.82·32-s − 4.54·34-s − 3.62·37-s − 12.0·38-s + 3.28·41-s − 9.35·43-s − 3.91·44-s + 7.22·46-s + 9.51·47-s − 0.695·49-s − 1.28·52-s + ⋯ |
| L(s) = 1 | − 1.25·2-s + 0.576·4-s − 0.948·7-s + 0.532·8-s − 1.02·11-s − 0.308·13-s + 1.19·14-s − 1.24·16-s + 0.621·17-s + 1.55·19-s + 1.28·22-s − 0.848·23-s + 0.387·26-s − 0.546·28-s + 0.351·29-s − 0.179·31-s + 1.02·32-s − 0.780·34-s − 0.595·37-s − 1.95·38-s + 0.512·41-s − 1.42·43-s − 0.590·44-s + 1.06·46-s + 1.38·47-s − 0.0994·49-s − 0.177·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6975 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 31 | \( 1 + T \) |
| good | 2 | \( 1 + 1.77T + 2T^{2} \) |
| 7 | \( 1 + 2.51T + 7T^{2} \) |
| 11 | \( 1 + 3.40T + 11T^{2} \) |
| 13 | \( 1 + 1.11T + 13T^{2} \) |
| 17 | \( 1 - 2.56T + 17T^{2} \) |
| 19 | \( 1 - 6.79T + 19T^{2} \) |
| 23 | \( 1 + 4.06T + 23T^{2} \) |
| 29 | \( 1 - 1.89T + 29T^{2} \) |
| 37 | \( 1 + 3.62T + 37T^{2} \) |
| 41 | \( 1 - 3.28T + 41T^{2} \) |
| 43 | \( 1 + 9.35T + 43T^{2} \) |
| 47 | \( 1 - 9.51T + 47T^{2} \) |
| 53 | \( 1 + 1.72T + 53T^{2} \) |
| 59 | \( 1 + 7.07T + 59T^{2} \) |
| 61 | \( 1 - 13.7T + 61T^{2} \) |
| 67 | \( 1 + 5.54T + 67T^{2} \) |
| 71 | \( 1 - 13.8T + 71T^{2} \) |
| 73 | \( 1 + 1.28T + 73T^{2} \) |
| 79 | \( 1 - 0.549T + 79T^{2} \) |
| 83 | \( 1 - 9.53T + 83T^{2} \) |
| 89 | \( 1 - 9.76T + 89T^{2} \) |
| 97 | \( 1 + 14.5T + 97T^{2} \) |
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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
−7.72647652821094589261896890640, −7.19639482958824633632301977421, −6.43005441513192205218470460936, −5.49586192190947850335265339701, −4.92458517174282335551491426252, −3.77712976239854673553419571525, −3.00399668292915946675946758191, −2.11192858363423729384440685632, −0.961068728629022901143324006103, 0,
0.961068728629022901143324006103, 2.11192858363423729384440685632, 3.00399668292915946675946758191, 3.77712976239854673553419571525, 4.92458517174282335551491426252, 5.49586192190947850335265339701, 6.43005441513192205218470460936, 7.19639482958824633632301977421, 7.72647652821094589261896890640
