| L(s) = 1 | − 2-s + 4-s + 5-s − 8-s − 10-s − 0.585·11-s + 16-s + 1.41·17-s + 2.82·19-s + 20-s + 0.585·22-s − 4.82·23-s + 25-s − 8.24·29-s − 5.07·31-s − 32-s − 1.41·34-s − 1.41·37-s − 2.82·38-s − 40-s − 8.82·41-s + 4.58·43-s − 0.585·44-s + 4.82·46-s − 9.07·47-s − 50-s + 9.31·53-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s + 0.447·5-s − 0.353·8-s − 0.316·10-s − 0.176·11-s + 0.250·16-s + 0.342·17-s + 0.648·19-s + 0.223·20-s + 0.124·22-s − 1.00·23-s + 0.200·25-s − 1.53·29-s − 0.910·31-s − 0.176·32-s − 0.242·34-s − 0.232·37-s − 0.458·38-s − 0.158·40-s − 1.37·41-s + 0.699·43-s − 0.0883·44-s + 0.711·46-s − 1.32·47-s − 0.141·50-s + 1.27·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4410 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4410 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 + 0.585T + 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 - 1.41T + 17T^{2} \) |
| 19 | \( 1 - 2.82T + 19T^{2} \) |
| 23 | \( 1 + 4.82T + 23T^{2} \) |
| 29 | \( 1 + 8.24T + 29T^{2} \) |
| 31 | \( 1 + 5.07T + 31T^{2} \) |
| 37 | \( 1 + 1.41T + 37T^{2} \) |
| 41 | \( 1 + 8.82T + 41T^{2} \) |
| 43 | \( 1 - 4.58T + 43T^{2} \) |
| 47 | \( 1 + 9.07T + 47T^{2} \) |
| 53 | \( 1 - 9.31T + 53T^{2} \) |
| 59 | \( 1 - 2.48T + 59T^{2} \) |
| 61 | \( 1 - 11.6T + 61T^{2} \) |
| 67 | \( 1 - 7.89T + 67T^{2} \) |
| 71 | \( 1 + 10.8T + 71T^{2} \) |
| 73 | \( 1 + 7.65T + 73T^{2} \) |
| 79 | \( 1 + 11.3T + 79T^{2} \) |
| 83 | \( 1 + 5.17T + 83T^{2} \) |
| 89 | \( 1 + 6.48T + 89T^{2} \) |
| 97 | \( 1 + 8.82T + 97T^{2} \) |
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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
−8.047510392675918253894920098590, −7.32287935039268121435515831255, −6.73445762050930757214098397557, −5.68119668602881275579526228020, −5.38276377125478138242689117389, −4.07872466930935751139230104927, −3.25084160218310597280720462982, −2.20972809059151474681798338423, −1.41713118953117451274512190717, 0,
1.41713118953117451274512190717, 2.20972809059151474681798338423, 3.25084160218310597280720462982, 4.07872466930935751139230104927, 5.38276377125478138242689117389, 5.68119668602881275579526228020, 6.73445762050930757214098397557, 7.32287935039268121435515831255, 8.047510392675918253894920098590
