| L(s) = 1 | − 2.17·2-s + 0.539·3-s + 2.70·4-s − 1.17·6-s − 4.87·7-s − 1.53·8-s − 2.70·9-s + 3.17·11-s + 1.46·12-s − 2.63·13-s + 10.5·14-s − 2.07·16-s + 5.87·18-s − 1.07·19-s − 2.63·21-s − 6.87·22-s + 5.21·23-s − 0.829·24-s + 5.70·26-s − 3.07·27-s − 13.2·28-s − 2.92·29-s + 4.09·31-s + 7.58·32-s + 1.70·33-s − 7.34·36-s − 5.26·37-s + ⋯ |
| L(s) = 1 | − 1.53·2-s + 0.311·3-s + 1.35·4-s − 0.477·6-s − 1.84·7-s − 0.544·8-s − 0.903·9-s + 0.955·11-s + 0.421·12-s − 0.729·13-s + 2.82·14-s − 0.519·16-s + 1.38·18-s − 0.247·19-s − 0.574·21-s − 1.46·22-s + 1.08·23-s − 0.169·24-s + 1.11·26-s − 0.592·27-s − 2.49·28-s − 0.542·29-s + 0.734·31-s + 1.34·32-s + 0.297·33-s − 1.22·36-s − 0.865·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 + 2.17T + 2T^{2} \) |
| 3 | \( 1 - 0.539T + 3T^{2} \) |
| 7 | \( 1 + 4.87T + 7T^{2} \) |
| 11 | \( 1 - 3.17T + 11T^{2} \) |
| 13 | \( 1 + 2.63T + 13T^{2} \) |
| 19 | \( 1 + 1.07T + 19T^{2} \) |
| 23 | \( 1 - 5.21T + 23T^{2} \) |
| 29 | \( 1 + 2.92T + 29T^{2} \) |
| 31 | \( 1 - 4.09T + 31T^{2} \) |
| 37 | \( 1 + 5.26T + 37T^{2} \) |
| 41 | \( 1 - 5.60T + 41T^{2} \) |
| 43 | \( 1 - 3.36T + 43T^{2} \) |
| 47 | \( 1 - 6.78T + 47T^{2} \) |
| 53 | \( 1 - 3.75T + 53T^{2} \) |
| 59 | \( 1 - 2.34T + 59T^{2} \) |
| 61 | \( 1 + 12.2T + 61T^{2} \) |
| 67 | \( 1 + 10.2T + 67T^{2} \) |
| 71 | \( 1 - 4.06T + 71T^{2} \) |
| 73 | \( 1 + 11.0T + 73T^{2} \) |
| 79 | \( 1 + 6.92T + 79T^{2} \) |
| 83 | \( 1 - 8.23T + 83T^{2} \) |
| 89 | \( 1 - 7.15T + 89T^{2} \) |
| 97 | \( 1 - 8.18T + 97T^{2} \) |
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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
−7.41914337412073888657133780959, −7.26214720382447107115710708858, −6.31134056501354947190763934497, −5.94971021946434980616038471932, −4.66922312541608202418863359511, −3.61964169751433426100513596243, −2.91102072948207525074856428167, −2.21732941116969033781629279994, −0.903258144481550607761495017929, 0,
0.903258144481550607761495017929, 2.21732941116969033781629279994, 2.91102072948207525074856428167, 3.61964169751433426100513596243, 4.66922312541608202418863359511, 5.94971021946434980616038471932, 6.31134056501354947190763934497, 7.26214720382447107115710708858, 7.41914337412073888657133780959
