L(s) = 1 | + 3-s − 3.41·5-s − 7-s + 9-s − 2.82·11-s + 5.65·13-s − 3.41·15-s + 2.24·17-s + 19-s − 21-s − 8.48·23-s + 6.65·25-s + 27-s + 10.2·29-s + 8.82·31-s − 2.82·33-s + 3.41·35-s − 6·37-s + 5.65·39-s − 8.82·41-s − 8.82·43-s − 3.41·45-s − 10.5·47-s + 49-s + 2.24·51-s − 5.07·53-s + 9.65·55-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 1.52·5-s − 0.377·7-s + 0.333·9-s − 0.852·11-s + 1.56·13-s − 0.881·15-s + 0.543·17-s + 0.229·19-s − 0.218·21-s − 1.76·23-s + 1.33·25-s + 0.192·27-s + 1.90·29-s + 1.58·31-s − 0.492·33-s + 0.577·35-s − 0.986·37-s + 0.905·39-s − 1.37·41-s − 1.34·43-s − 0.508·45-s − 1.54·47-s + 0.142·49-s + 0.314·51-s − 0.696·53-s + 1.30·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3192 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 19 | \( 1 - T \) |
good | 5 | \( 1 + 3.41T + 5T^{2} \) |
| 11 | \( 1 + 2.82T + 11T^{2} \) |
| 13 | \( 1 - 5.65T + 13T^{2} \) |
| 17 | \( 1 - 2.24T + 17T^{2} \) |
| 23 | \( 1 + 8.48T + 23T^{2} \) |
| 29 | \( 1 - 10.2T + 29T^{2} \) |
| 31 | \( 1 - 8.82T + 31T^{2} \) |
| 37 | \( 1 + 6T + 37T^{2} \) |
| 41 | \( 1 + 8.82T + 41T^{2} \) |
| 43 | \( 1 + 8.82T + 43T^{2} \) |
| 47 | \( 1 + 10.5T + 47T^{2} \) |
| 53 | \( 1 + 5.07T + 53T^{2} \) |
| 59 | \( 1 + 2.34T + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 + 9.17T + 67T^{2} \) |
| 71 | \( 1 - 12.7T + 71T^{2} \) |
| 73 | \( 1 + 8.82T + 73T^{2} \) |
| 79 | \( 1 - 2.34T + 79T^{2} \) |
| 83 | \( 1 + 12.2T + 83T^{2} \) |
| 89 | \( 1 + 16.8T + 89T^{2} \) |
| 97 | \( 1 + 0.343T + 97T^{2} \) |
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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
−8.274935678345561594433718390332, −7.903796292839556892021560512037, −6.83477816890128458220995817469, −6.23352844012666151409034838664, −5.03506044716846449582163155312, −4.20547437352001516235749198810, −3.44354079713731384784731494977, −2.93877897115368570253503994234, −1.40567775873301748409140799060, 0,
1.40567775873301748409140799060, 2.93877897115368570253503994234, 3.44354079713731384784731494977, 4.20547437352001516235749198810, 5.03506044716846449582163155312, 6.23352844012666151409034838664, 6.83477816890128458220995817469, 7.903796292839556892021560512037, 8.274935678345561594433718390332