L(s) = 1 | + 0.414·3-s + 5-s + 2.82·7-s − 2.82·9-s − 2.41·11-s − 1.82·13-s + 0.414·15-s − 4.82·17-s − 6·19-s + 1.17·21-s − 7.65·23-s − 4·25-s − 2.41·27-s − 29-s − 4.07·31-s − 0.999·33-s + 2.82·35-s + 4·37-s − 0.757·39-s + 12.4·41-s − 6.41·43-s − 2.82·45-s + 5.24·47-s + 1.00·49-s − 1.99·51-s + 7.48·53-s − 2.41·55-s + ⋯ |
L(s) = 1 | + 0.239·3-s + 0.447·5-s + 1.06·7-s − 0.942·9-s − 0.727·11-s − 0.507·13-s + 0.106·15-s − 1.17·17-s − 1.37·19-s + 0.255·21-s − 1.59·23-s − 0.800·25-s − 0.464·27-s − 0.185·29-s − 0.731·31-s − 0.174·33-s + 0.478·35-s + 0.657·37-s − 0.121·39-s + 1.94·41-s − 0.978·43-s − 0.421·45-s + 0.764·47-s + 0.142·49-s − 0.280·51-s + 1.02·53-s − 0.325·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1856 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1856 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 29 | \( 1 + T \) |
good | 3 | \( 1 - 0.414T + 3T^{2} \) |
| 5 | \( 1 - T + 5T^{2} \) |
| 7 | \( 1 - 2.82T + 7T^{2} \) |
| 11 | \( 1 + 2.41T + 11T^{2} \) |
| 13 | \( 1 + 1.82T + 13T^{2} \) |
| 17 | \( 1 + 4.82T + 17T^{2} \) |
| 19 | \( 1 + 6T + 19T^{2} \) |
| 23 | \( 1 + 7.65T + 23T^{2} \) |
| 31 | \( 1 + 4.07T + 31T^{2} \) |
| 37 | \( 1 - 4T + 37T^{2} \) |
| 41 | \( 1 - 12.4T + 41T^{2} \) |
| 43 | \( 1 + 6.41T + 43T^{2} \) |
| 47 | \( 1 - 5.24T + 47T^{2} \) |
| 53 | \( 1 - 7.48T + 53T^{2} \) |
| 59 | \( 1 + 7.65T + 59T^{2} \) |
| 61 | \( 1 + 0.828T + 61T^{2} \) |
| 67 | \( 1 - 5.65T + 67T^{2} \) |
| 71 | \( 1 + 3.17T + 71T^{2} \) |
| 73 | \( 1 - 4T + 73T^{2} \) |
| 79 | \( 1 - 0.414T + 79T^{2} \) |
| 83 | \( 1 - 3.65T + 83T^{2} \) |
| 89 | \( 1 - 4.48T + 89T^{2} \) |
| 97 | \( 1 + 12.4T + 97T^{2} \) |
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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
−8.752224356421627575261631364288, −8.098372663257254115582378366429, −7.52474121636763809768949184922, −6.26595221991533593513321338243, −5.68489733783248772871386675851, −4.73518046796162665483122124709, −3.94689135092261173088975092856, −2.39509171048494079004273609655, −2.06898729966108955556495974338, 0,
2.06898729966108955556495974338, 2.39509171048494079004273609655, 3.94689135092261173088975092856, 4.73518046796162665483122124709, 5.68489733783248772871386675851, 6.26595221991533593513321338243, 7.52474121636763809768949184922, 8.098372663257254115582378366429, 8.752224356421627575261631364288