| L(s) = 1 | + 2-s + 3-s + 4-s + 6-s + 8-s + 9-s + 12-s − 4·13-s + 16-s + 3·17-s + 18-s − 2·19-s − 3·23-s + 24-s − 4·26-s + 27-s + 31-s + 32-s + 3·34-s + 36-s + 10·37-s − 2·38-s − 4·39-s + 9·41-s + 10·43-s − 3·46-s − 3·47-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s + 0.353·8-s + 1/3·9-s + 0.288·12-s − 1.10·13-s + 1/4·16-s + 0.727·17-s + 0.235·18-s − 0.458·19-s − 0.625·23-s + 0.204·24-s − 0.784·26-s + 0.192·27-s + 0.179·31-s + 0.176·32-s + 0.514·34-s + 1/6·36-s + 1.64·37-s − 0.324·38-s − 0.640·39-s + 1.40·41-s + 1.52·43-s − 0.442·46-s − 0.437·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.072426278\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.072426278\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 3 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 - 11 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 15 T + p T^{2} \) |
| 97 | \( 1 + 7 T + p T^{2} \) |
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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
−7.75187134175619295120059178632, −7.33862919231070881505323017399, −6.41937999388886302299885425167, −5.80116866268330422631854756048, −4.96954554347682342117052352497, −4.27728364915519111045619457716, −3.63995850192515348454197204945, −2.62146741432360024211328660371, −2.22158978001962975092515124566, −0.887944806429440248023526470993,
0.887944806429440248023526470993, 2.22158978001962975092515124566, 2.62146741432360024211328660371, 3.63995850192515348454197204945, 4.27728364915519111045619457716, 4.96954554347682342117052352497, 5.80116866268330422631854756048, 6.41937999388886302299885425167, 7.33862919231070881505323017399, 7.75187134175619295120059178632
