L(s) = 1 | − 2-s + 3-s + 4-s + 1.30·5-s − 6-s + 2.83·7-s − 8-s + 9-s − 1.30·10-s + 5.04·11-s + 12-s − 13-s − 2.83·14-s + 1.30·15-s + 16-s + 2.94·17-s − 18-s + 3.34·19-s + 1.30·20-s + 2.83·21-s − 5.04·22-s − 0.726·23-s − 24-s − 3.28·25-s + 26-s + 27-s + 2.83·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.585·5-s − 0.408·6-s + 1.07·7-s − 0.353·8-s + 0.333·9-s − 0.413·10-s + 1.51·11-s + 0.288·12-s − 0.277·13-s − 0.758·14-s + 0.337·15-s + 0.250·16-s + 0.714·17-s − 0.235·18-s + 0.767·19-s + 0.292·20-s + 0.619·21-s − 1.07·22-s − 0.151·23-s − 0.204·24-s − 0.657·25-s + 0.196·26-s + 0.192·27-s + 0.536·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.916255303\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.916255303\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| 103 | \( 1 + T \) |
good | 5 | \( 1 - 1.30T + 5T^{2} \) |
| 7 | \( 1 - 2.83T + 7T^{2} \) |
| 11 | \( 1 - 5.04T + 11T^{2} \) |
| 17 | \( 1 - 2.94T + 17T^{2} \) |
| 19 | \( 1 - 3.34T + 19T^{2} \) |
| 23 | \( 1 + 0.726T + 23T^{2} \) |
| 29 | \( 1 + 5.62T + 29T^{2} \) |
| 31 | \( 1 + 2.12T + 31T^{2} \) |
| 37 | \( 1 - 1.95T + 37T^{2} \) |
| 41 | \( 1 - 2.36T + 41T^{2} \) |
| 43 | \( 1 - 4.27T + 43T^{2} \) |
| 47 | \( 1 + 12.5T + 47T^{2} \) |
| 53 | \( 1 - 10.8T + 53T^{2} \) |
| 59 | \( 1 - 9.82T + 59T^{2} \) |
| 61 | \( 1 - 11.5T + 61T^{2} \) |
| 67 | \( 1 - 3.16T + 67T^{2} \) |
| 71 | \( 1 - 5.90T + 71T^{2} \) |
| 73 | \( 1 + 8.75T + 73T^{2} \) |
| 79 | \( 1 - 12.3T + 79T^{2} \) |
| 83 | \( 1 + 9.61T + 83T^{2} \) |
| 89 | \( 1 - 15.1T + 89T^{2} \) |
| 97 | \( 1 + 4.99T + 97T^{2} \) |
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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
−7.952523088044615061404477253429, −7.28866423530966529131074689450, −6.66851784134939256446743338561, −5.75788380676650863240398295911, −5.18686197840959254437060639032, −4.09363961185404648443446337728, −3.51063885237823249621083382518, −2.34750541320135605085947071667, −1.69825698498526284559393444895, −1.00289729427175656740876183143,
1.00289729427175656740876183143, 1.69825698498526284559393444895, 2.34750541320135605085947071667, 3.51063885237823249621083382518, 4.09363961185404648443446337728, 5.18686197840959254437060639032, 5.75788380676650863240398295911, 6.66851784134939256446743338561, 7.28866423530966529131074689450, 7.952523088044615061404477253429