L(s) = 1 | − 2-s + 1.92·3-s + 4-s + 3.36·5-s − 1.92·6-s − 0.755·7-s − 8-s + 0.710·9-s − 3.36·10-s − 3.68·11-s + 1.92·12-s − 2.93·13-s + 0.755·14-s + 6.48·15-s + 16-s − 4.39·17-s − 0.710·18-s − 3.42·19-s + 3.36·20-s − 1.45·21-s + 3.68·22-s − 7.53·23-s − 1.92·24-s + 6.32·25-s + 2.93·26-s − 4.41·27-s − 0.755·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1.11·3-s + 0.5·4-s + 1.50·5-s − 0.786·6-s − 0.285·7-s − 0.353·8-s + 0.236·9-s − 1.06·10-s − 1.11·11-s + 0.556·12-s − 0.814·13-s + 0.201·14-s + 1.67·15-s + 0.250·16-s − 1.06·17-s − 0.167·18-s − 0.784·19-s + 0.752·20-s − 0.317·21-s + 0.786·22-s − 1.57·23-s − 0.393·24-s + 1.26·25-s + 0.575·26-s − 0.848·27-s − 0.142·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 2017 | \( 1 + T \) |
good | 3 | \( 1 - 1.92T + 3T^{2} \) |
| 5 | \( 1 - 3.36T + 5T^{2} \) |
| 7 | \( 1 + 0.755T + 7T^{2} \) |
| 11 | \( 1 + 3.68T + 11T^{2} \) |
| 13 | \( 1 + 2.93T + 13T^{2} \) |
| 17 | \( 1 + 4.39T + 17T^{2} \) |
| 19 | \( 1 + 3.42T + 19T^{2} \) |
| 23 | \( 1 + 7.53T + 23T^{2} \) |
| 29 | \( 1 - 7.45T + 29T^{2} \) |
| 31 | \( 1 + 6.01T + 31T^{2} \) |
| 37 | \( 1 - 6.19T + 37T^{2} \) |
| 41 | \( 1 - 2.70T + 41T^{2} \) |
| 43 | \( 1 + 3.24T + 43T^{2} \) |
| 47 | \( 1 + 5.46T + 47T^{2} \) |
| 53 | \( 1 - 12.1T + 53T^{2} \) |
| 59 | \( 1 - 0.876T + 59T^{2} \) |
| 61 | \( 1 - 7.76T + 61T^{2} \) |
| 67 | \( 1 + 10.6T + 67T^{2} \) |
| 71 | \( 1 + 12.2T + 71T^{2} \) |
| 73 | \( 1 - 12.7T + 73T^{2} \) |
| 79 | \( 1 + 2.08T + 79T^{2} \) |
| 83 | \( 1 + 9.40T + 83T^{2} \) |
| 89 | \( 1 - 6.06T + 89T^{2} \) |
| 97 | \( 1 + 0.132T + 97T^{2} \) |
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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
−8.272073139228727645297403830982, −7.57004262579367463306541446847, −6.65831515517896397407879391713, −6.01938569789087984828481236215, −5.23212049298637158437310234518, −4.16268226715447929681859805318, −2.84157105018865433145586463947, −2.39235824906665390363824345765, −1.83384336292064948385083646774, 0,
1.83384336292064948385083646774, 2.39235824906665390363824345765, 2.84157105018865433145586463947, 4.16268226715447929681859805318, 5.23212049298637158437310234518, 6.01938569789087984828481236215, 6.65831515517896397407879391713, 7.57004262579367463306541446847, 8.272073139228727645297403830982