L(s) = 1 | − 1.73·2-s − 2.59·3-s + 0.998·4-s − 5-s + 4.49·6-s − 7-s + 1.73·8-s + 3.73·9-s + 1.73·10-s − 5.80·11-s − 2.58·12-s − 4.81·13-s + 1.73·14-s + 2.59·15-s − 4.99·16-s − 6.09·17-s − 6.46·18-s − 1.57·19-s − 0.998·20-s + 2.59·21-s + 10.0·22-s + 3.03·23-s − 4.50·24-s + 25-s + 8.33·26-s − 1.89·27-s − 0.998·28-s + ⋯ |
L(s) = 1 | − 1.22·2-s − 1.49·3-s + 0.499·4-s − 0.447·5-s + 1.83·6-s − 0.377·7-s + 0.613·8-s + 1.24·9-s + 0.547·10-s − 1.74·11-s − 0.747·12-s − 1.33·13-s + 0.462·14-s + 0.669·15-s − 1.24·16-s − 1.47·17-s − 1.52·18-s − 0.362·19-s − 0.223·20-s + 0.566·21-s + 2.14·22-s + 0.631·23-s − 0.918·24-s + 0.200·25-s + 1.63·26-s − 0.365·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 229 | \( 1 + T \) |
good | 2 | \( 1 + 1.73T + 2T^{2} \) |
| 3 | \( 1 + 2.59T + 3T^{2} \) |
| 11 | \( 1 + 5.80T + 11T^{2} \) |
| 13 | \( 1 + 4.81T + 13T^{2} \) |
| 17 | \( 1 + 6.09T + 17T^{2} \) |
| 19 | \( 1 + 1.57T + 19T^{2} \) |
| 23 | \( 1 - 3.03T + 23T^{2} \) |
| 29 | \( 1 + 1.20T + 29T^{2} \) |
| 31 | \( 1 + 6.38T + 31T^{2} \) |
| 37 | \( 1 - 6.93T + 37T^{2} \) |
| 41 | \( 1 - 0.935T + 41T^{2} \) |
| 43 | \( 1 + 2.81T + 43T^{2} \) |
| 47 | \( 1 + 7.57T + 47T^{2} \) |
| 53 | \( 1 + 1.39T + 53T^{2} \) |
| 59 | \( 1 - 12.5T + 59T^{2} \) |
| 61 | \( 1 - 2.64T + 61T^{2} \) |
| 67 | \( 1 - 0.00740T + 67T^{2} \) |
| 71 | \( 1 + 1.56T + 71T^{2} \) |
| 73 | \( 1 + 16.2T + 73T^{2} \) |
| 79 | \( 1 - 9.63T + 79T^{2} \) |
| 83 | \( 1 + 4.78T + 83T^{2} \) |
| 89 | \( 1 - 9.59T + 89T^{2} \) |
| 97 | \( 1 + 14.1T + 97T^{2} \) |
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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
−7.39075138223288752288414338679, −7.04739514645094886105725347410, −6.27201499984906517804737251310, −5.28740347465368884426289517919, −4.88755709036447697802827864940, −4.22208522147201605729179681539, −2.78370290743622181755195168497, −1.98459425214462918407996683821, −0.53530571312770429468022735048, 0,
0.53530571312770429468022735048, 1.98459425214462918407996683821, 2.78370290743622181755195168497, 4.22208522147201605729179681539, 4.88755709036447697802827864940, 5.28740347465368884426289517919, 6.27201499984906517804737251310, 7.04739514645094886105725347410, 7.39075138223288752288414338679