| L(s) = 1 | − 2.47·2-s + 4.11·4-s + 2.58·5-s − 7-s − 5.22·8-s − 6.39·10-s + 11-s − 5.87·13-s + 2.47·14-s + 4.70·16-s − 7.51·17-s − 2.35·19-s + 10.6·20-s − 2.47·22-s − 6.94·23-s + 1.69·25-s + 14.5·26-s − 4.11·28-s + 5.87·29-s − 3.66·31-s − 1.16·32-s + 18.5·34-s − 2.58·35-s + 3.30·37-s + 5.83·38-s − 13.5·40-s − 5.28·41-s + ⋯ |
| L(s) = 1 | − 1.74·2-s + 2.05·4-s + 1.15·5-s − 0.377·7-s − 1.84·8-s − 2.02·10-s + 0.301·11-s − 1.62·13-s + 0.660·14-s + 1.17·16-s − 1.82·17-s − 0.540·19-s + 2.38·20-s − 0.527·22-s − 1.44·23-s + 0.339·25-s + 2.84·26-s − 0.777·28-s + 1.09·29-s − 0.657·31-s − 0.206·32-s + 3.18·34-s − 0.437·35-s + 0.543·37-s + 0.945·38-s − 2.13·40-s − 0.825·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| good | 2 | \( 1 + 2.47T + 2T^{2} \) |
| 5 | \( 1 - 2.58T + 5T^{2} \) |
| 13 | \( 1 + 5.87T + 13T^{2} \) |
| 17 | \( 1 + 7.51T + 17T^{2} \) |
| 19 | \( 1 + 2.35T + 19T^{2} \) |
| 23 | \( 1 + 6.94T + 23T^{2} \) |
| 29 | \( 1 - 5.87T + 29T^{2} \) |
| 31 | \( 1 + 3.66T + 31T^{2} \) |
| 37 | \( 1 - 3.30T + 37T^{2} \) |
| 41 | \( 1 + 5.28T + 41T^{2} \) |
| 43 | \( 1 - 7.40T + 43T^{2} \) |
| 47 | \( 1 + 7.53T + 47T^{2} \) |
| 53 | \( 1 - 4.22T + 53T^{2} \) |
| 59 | \( 1 - 0.926T + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 + 10.1T + 67T^{2} \) |
| 71 | \( 1 + 4.45T + 71T^{2} \) |
| 73 | \( 1 + 2.12T + 73T^{2} \) |
| 79 | \( 1 + 4.45T + 79T^{2} \) |
| 83 | \( 1 - 10.6T + 83T^{2} \) |
| 89 | \( 1 + 15.8T + 89T^{2} \) |
| 97 | \( 1 - 10.1T + 97T^{2} \) |
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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
−9.960107807087348120975795329477, −9.250753148388984556092838574682, −8.610533503782056568724918333153, −7.52134834263108139901158957655, −6.67504972162371151591638790190, −6.03676835054156400186328809877, −4.55212610203629738141904396619, −2.52690372605223738358154073362, −1.90906479945174843608366595062, 0,
1.90906479945174843608366595062, 2.52690372605223738358154073362, 4.55212610203629738141904396619, 6.03676835054156400186328809877, 6.67504972162371151591638790190, 7.52134834263108139901158957655, 8.610533503782056568724918333153, 9.250753148388984556092838574682, 9.960107807087348120975795329477
