L(s) = 1 | − 2.02·2-s − 1.81·3-s + 2.09·4-s + 3.66·6-s − 3.10·7-s − 0.192·8-s + 0.280·9-s + 1.54·11-s − 3.79·12-s + 1.50·13-s + 6.28·14-s − 3.80·16-s − 3.57·17-s − 0.568·18-s + 4.32·19-s + 5.62·21-s − 3.12·22-s + 2.78·23-s + 0.347·24-s − 3.04·26-s + 4.92·27-s − 6.50·28-s − 7.83·29-s + 4.16·31-s + 8.07·32-s − 2.79·33-s + 7.23·34-s + ⋯ |
L(s) = 1 | − 1.43·2-s − 1.04·3-s + 1.04·4-s + 1.49·6-s − 1.17·7-s − 0.0678·8-s + 0.0936·9-s + 0.465·11-s − 1.09·12-s + 0.417·13-s + 1.68·14-s − 0.950·16-s − 0.867·17-s − 0.133·18-s + 0.992·19-s + 1.22·21-s − 0.665·22-s + 0.581·23-s + 0.0709·24-s − 0.597·26-s + 0.947·27-s − 1.23·28-s − 1.45·29-s + 0.748·31-s + 1.42·32-s − 0.486·33-s + 1.24·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 47 | \( 1 - T \) |
good | 2 | \( 1 + 2.02T + 2T^{2} \) |
| 3 | \( 1 + 1.81T + 3T^{2} \) |
| 7 | \( 1 + 3.10T + 7T^{2} \) |
| 11 | \( 1 - 1.54T + 11T^{2} \) |
| 13 | \( 1 - 1.50T + 13T^{2} \) |
| 17 | \( 1 + 3.57T + 17T^{2} \) |
| 19 | \( 1 - 4.32T + 19T^{2} \) |
| 23 | \( 1 - 2.78T + 23T^{2} \) |
| 29 | \( 1 + 7.83T + 29T^{2} \) |
| 31 | \( 1 - 4.16T + 31T^{2} \) |
| 37 | \( 1 - 5.83T + 37T^{2} \) |
| 41 | \( 1 - 9.25T + 41T^{2} \) |
| 43 | \( 1 + 9.48T + 43T^{2} \) |
| 53 | \( 1 - 12.3T + 53T^{2} \) |
| 59 | \( 1 + 2.15T + 59T^{2} \) |
| 61 | \( 1 + 11.8T + 61T^{2} \) |
| 67 | \( 1 - 6.15T + 67T^{2} \) |
| 71 | \( 1 - 2.47T + 71T^{2} \) |
| 73 | \( 1 + 2.11T + 73T^{2} \) |
| 79 | \( 1 + 3.60T + 79T^{2} \) |
| 83 | \( 1 + 1.39T + 83T^{2} \) |
| 89 | \( 1 - 3.13T + 89T^{2} \) |
| 97 | \( 1 + 13.6T + 97T^{2} \) |
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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
−9.375942468118509827453286569648, −8.837101307435466508332806556916, −7.75690839909176102789208007983, −6.83863447797492328676641986636, −6.32553378921201778665516754722, −5.36599218877642289237057891174, −4.10612267179395052538979573813, −2.75615940225402917011772275072, −1.14478571735400017129340012443, 0,
1.14478571735400017129340012443, 2.75615940225402917011772275072, 4.10612267179395052538979573813, 5.36599218877642289237057891174, 6.32553378921201778665516754722, 6.83863447797492328676641986636, 7.75690839909176102789208007983, 8.837101307435466508332806556916, 9.375942468118509827453286569648