| L(s) = 1 | − 2-s + 3-s + 4-s + 1.82·5-s − 6-s + 1.67·7-s − 8-s + 9-s − 1.82·10-s + 2.01·11-s + 12-s − 4.91·13-s − 1.67·14-s + 1.82·15-s + 16-s − 6.76·17-s − 18-s + 19-s + 1.82·20-s + 1.67·21-s − 2.01·22-s − 3.81·23-s − 24-s − 1.65·25-s + 4.91·26-s + 27-s + 1.67·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.818·5-s − 0.408·6-s + 0.634·7-s − 0.353·8-s + 0.333·9-s − 0.578·10-s + 0.607·11-s + 0.288·12-s − 1.36·13-s − 0.448·14-s + 0.472·15-s + 0.250·16-s − 1.64·17-s − 0.235·18-s + 0.229·19-s + 0.409·20-s + 0.366·21-s − 0.429·22-s − 0.795·23-s − 0.204·24-s − 0.330·25-s + 0.964·26-s + 0.192·27-s + 0.317·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6042 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6042 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| 53 | \( 1 + T \) |
| good | 5 | \( 1 - 1.82T + 5T^{2} \) |
| 7 | \( 1 - 1.67T + 7T^{2} \) |
| 11 | \( 1 - 2.01T + 11T^{2} \) |
| 13 | \( 1 + 4.91T + 13T^{2} \) |
| 17 | \( 1 + 6.76T + 17T^{2} \) |
| 23 | \( 1 + 3.81T + 23T^{2} \) |
| 29 | \( 1 - 2.88T + 29T^{2} \) |
| 31 | \( 1 + 4.78T + 31T^{2} \) |
| 37 | \( 1 + 7.07T + 37T^{2} \) |
| 41 | \( 1 + 9.22T + 41T^{2} \) |
| 43 | \( 1 - 5.31T + 43T^{2} \) |
| 47 | \( 1 + 0.326T + 47T^{2} \) |
| 59 | \( 1 + 1.93T + 59T^{2} \) |
| 61 | \( 1 + 2.19T + 61T^{2} \) |
| 67 | \( 1 - 1.21T + 67T^{2} \) |
| 71 | \( 1 - 5.02T + 71T^{2} \) |
| 73 | \( 1 + 14.3T + 73T^{2} \) |
| 79 | \( 1 + 5.68T + 79T^{2} \) |
| 83 | \( 1 - 3.63T + 83T^{2} \) |
| 89 | \( 1 + 11.9T + 89T^{2} \) |
| 97 | \( 1 - 14.3T + 97T^{2} \) |
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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
−7.83800643889814200980358936164, −7.04895095966361394455233501737, −6.58821355533268404126246005134, −5.63035239970630253039546810259, −4.82682151502670615832692726782, −4.05406428718801828741395832245, −2.90717347282616312615373689331, −2.03151896143531308656919041524, −1.66423129650310354044017395760, 0,
1.66423129650310354044017395760, 2.03151896143531308656919041524, 2.90717347282616312615373689331, 4.05406428718801828741395832245, 4.82682151502670615832692726782, 5.63035239970630253039546810259, 6.58821355533268404126246005134, 7.04895095966361394455233501737, 7.83800643889814200980358936164
