L(s) = 1 | − 13.2·2-s − 162.·3-s − 337.·4-s − 1.69e3·5-s + 2.15e3·6-s + 1.04e4·7-s + 1.12e4·8-s + 6.74e3·9-s + 2.23e4·10-s − 2.79e4·11-s + 5.47e4·12-s − 1.47e5·13-s − 1.37e5·14-s + 2.74e5·15-s + 2.40e4·16-s − 8.92e4·18-s − 5.05e5·19-s + 5.69e5·20-s − 1.69e6·21-s + 3.69e5·22-s − 1.59e6·23-s − 1.82e6·24-s + 9.06e5·25-s + 1.95e6·26-s + 2.10e6·27-s − 3.51e6·28-s − 3.23e6·29-s + ⋯ |
L(s) = 1 | − 0.584·2-s − 1.15·3-s − 0.658·4-s − 1.20·5-s + 0.677·6-s + 1.64·7-s + 0.969·8-s + 0.342·9-s + 0.707·10-s − 0.574·11-s + 0.762·12-s − 1.43·13-s − 0.959·14-s + 1.40·15-s + 0.0915·16-s − 0.200·18-s − 0.890·19-s + 0.796·20-s − 1.90·21-s + 0.335·22-s − 1.18·23-s − 1.12·24-s + 0.463·25-s + 0.837·26-s + 0.761·27-s − 1.08·28-s − 0.849·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(5)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{11}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 17 | \( 1 \) |
good | 2 | \( 1 + 13.2T + 512T^{2} \) |
| 3 | \( 1 + 162.T + 1.96e4T^{2} \) |
| 5 | \( 1 + 1.69e3T + 1.95e6T^{2} \) |
| 7 | \( 1 - 1.04e4T + 4.03e7T^{2} \) |
| 11 | \( 1 + 2.79e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + 1.47e5T + 1.06e10T^{2} \) |
| 19 | \( 1 + 5.05e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 1.59e6T + 1.80e12T^{2} \) |
| 29 | \( 1 + 3.23e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 5.16e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 1.45e7T + 1.29e14T^{2} \) |
| 41 | \( 1 + 3.32e7T + 3.27e14T^{2} \) |
| 43 | \( 1 - 2.69e6T + 5.02e14T^{2} \) |
| 47 | \( 1 - 9.23e6T + 1.11e15T^{2} \) |
| 53 | \( 1 - 6.19e7T + 3.29e15T^{2} \) |
| 59 | \( 1 - 9.85e7T + 8.66e15T^{2} \) |
| 61 | \( 1 - 2.07e8T + 1.16e16T^{2} \) |
| 67 | \( 1 - 5.69e7T + 2.72e16T^{2} \) |
| 71 | \( 1 + 1.84e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + 3.74e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 1.08e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 4.94e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 1.69e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 7.68e8T + 7.60e17T^{2} \) |
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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
−10.00613201393215790264928471905, −8.460351824069947243961719069924, −8.007687376601129591216252470703, −7.17906989622370291773047520294, −5.51355698825878247698871476412, −4.75068316923247904322639725617, −4.15390101913796369041619645403, −2.11100414589467856448994549255, −0.70617919101925379681968462268, 0,
0.70617919101925379681968462268, 2.11100414589467856448994549255, 4.15390101913796369041619645403, 4.75068316923247904322639725617, 5.51355698825878247698871476412, 7.17906989622370291773047520294, 8.007687376601129591216252470703, 8.460351824069947243961719069924, 10.00613201393215790264928471905