| L(s) = 1 | − 1.73·2-s − 3.39·3-s + 1.01·4-s + 2.86·5-s + 5.89·6-s + 0.942·7-s + 1.71·8-s + 8.53·9-s − 4.96·10-s − 3.33·11-s − 3.43·12-s + 1.74·13-s − 1.63·14-s − 9.71·15-s − 4.99·16-s − 3.59·17-s − 14.8·18-s − 1.93·19-s + 2.89·20-s − 3.20·21-s + 5.78·22-s + 2.66·23-s − 5.82·24-s + 3.18·25-s − 3.03·26-s − 18.8·27-s + 0.953·28-s + ⋯ |
| L(s) = 1 | − 1.22·2-s − 1.96·3-s + 0.505·4-s + 1.27·5-s + 2.40·6-s + 0.356·7-s + 0.606·8-s + 2.84·9-s − 1.56·10-s − 1.00·11-s − 0.992·12-s + 0.484·13-s − 0.437·14-s − 2.50·15-s − 1.24·16-s − 0.871·17-s − 3.49·18-s − 0.445·19-s + 0.647·20-s − 0.698·21-s + 1.23·22-s + 0.554·23-s − 1.18·24-s + 0.636·25-s − 0.594·26-s − 3.61·27-s + 0.180·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9409 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9409 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 97 | \( 1 \) |
| good | 2 | \( 1 + 1.73T + 2T^{2} \) |
| 3 | \( 1 + 3.39T + 3T^{2} \) |
| 5 | \( 1 - 2.86T + 5T^{2} \) |
| 7 | \( 1 - 0.942T + 7T^{2} \) |
| 11 | \( 1 + 3.33T + 11T^{2} \) |
| 13 | \( 1 - 1.74T + 13T^{2} \) |
| 17 | \( 1 + 3.59T + 17T^{2} \) |
| 19 | \( 1 + 1.93T + 19T^{2} \) |
| 23 | \( 1 - 2.66T + 23T^{2} \) |
| 29 | \( 1 - 7.36T + 29T^{2} \) |
| 31 | \( 1 - 3.74T + 31T^{2} \) |
| 37 | \( 1 + 0.721T + 37T^{2} \) |
| 41 | \( 1 + 5.04T + 41T^{2} \) |
| 43 | \( 1 - 6.72T + 43T^{2} \) |
| 47 | \( 1 - 7.87T + 47T^{2} \) |
| 53 | \( 1 - 2.72T + 53T^{2} \) |
| 59 | \( 1 + 7.83T + 59T^{2} \) |
| 61 | \( 1 - 5.40T + 61T^{2} \) |
| 67 | \( 1 + 8.28T + 67T^{2} \) |
| 71 | \( 1 + 12.6T + 71T^{2} \) |
| 73 | \( 1 + 3.27T + 73T^{2} \) |
| 79 | \( 1 - 0.714T + 79T^{2} \) |
| 83 | \( 1 + 4.45T + 83T^{2} \) |
| 89 | \( 1 - 15.6T + 89T^{2} \) |
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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
−7.24839747543922078260918155259, −6.62090094029964204461301860703, −6.10370979508301383204567840032, −5.42824025896839038173721910186, −4.78256321945592672524366798933, −4.30589131599622286722947316411, −2.53935065682440611888056256389, −1.66004350889861485453704256108, −0.998838410408404415922822441874, 0,
0.998838410408404415922822441874, 1.66004350889861485453704256108, 2.53935065682440611888056256389, 4.30589131599622286722947316411, 4.78256321945592672524366798933, 5.42824025896839038173721910186, 6.10370979508301383204567840032, 6.62090094029964204461301860703, 7.24839747543922078260918155259
