| L(s) = 1 | − 3.29·5-s + 7-s + 11-s + 4.87·13-s − 0.426·17-s − 1.29·19-s + 5.87·25-s + 0.870·29-s − 4.16·31-s − 3.29·35-s − 0.870·37-s − 6.16·41-s + 4·43-s + 7.89·47-s + 49-s + 4.59·53-s − 3.29·55-s − 2.87·59-s − 10.3·61-s − 16.0·65-s − 0.276·67-s − 3.14·71-s + 13.0·73-s + 77-s + 8.16·83-s + 1.40·85-s + 6·89-s + ⋯ |
| L(s) = 1 | − 1.47·5-s + 0.377·7-s + 0.301·11-s + 1.35·13-s − 0.103·17-s − 0.297·19-s + 1.17·25-s + 0.161·29-s − 0.748·31-s − 0.557·35-s − 0.143·37-s − 0.963·41-s + 0.609·43-s + 1.15·47-s + 0.142·49-s + 0.631·53-s − 0.444·55-s − 0.373·59-s − 1.32·61-s − 1.99·65-s − 0.0337·67-s − 0.373·71-s + 1.52·73-s + 0.113·77-s + 0.896·83-s + 0.152·85-s + 0.635·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5544 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5544 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.444192829\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.444192829\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| good | 5 | \( 1 + 3.29T + 5T^{2} \) |
| 13 | \( 1 - 4.87T + 13T^{2} \) |
| 17 | \( 1 + 0.426T + 17T^{2} \) |
| 19 | \( 1 + 1.29T + 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 - 0.870T + 29T^{2} \) |
| 31 | \( 1 + 4.16T + 31T^{2} \) |
| 37 | \( 1 + 0.870T + 37T^{2} \) |
| 41 | \( 1 + 6.16T + 41T^{2} \) |
| 43 | \( 1 - 4T + 43T^{2} \) |
| 47 | \( 1 - 7.89T + 47T^{2} \) |
| 53 | \( 1 - 4.59T + 53T^{2} \) |
| 59 | \( 1 + 2.87T + 59T^{2} \) |
| 61 | \( 1 + 10.3T + 61T^{2} \) |
| 67 | \( 1 + 0.276T + 67T^{2} \) |
| 71 | \( 1 + 3.14T + 71T^{2} \) |
| 73 | \( 1 - 13.0T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 - 8.16T + 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 + 16.9T + 97T^{2} \) |
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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
−8.147948696968839480458861635915, −7.53763261773849569193258241425, −6.83573499745864754685168385372, −6.06568019040535528105480259463, −5.18190426360297525079788394293, −4.23107287089034196540423280791, −3.84338149593458821194567883471, −3.04462243954489068329085537133, −1.74847897514992250508892041809, −0.65818620640322108288765660524,
0.65818620640322108288765660524, 1.74847897514992250508892041809, 3.04462243954489068329085537133, 3.84338149593458821194567883471, 4.23107287089034196540423280791, 5.18190426360297525079788394293, 6.06568019040535528105480259463, 6.83573499745864754685168385372, 7.53763261773849569193258241425, 8.147948696968839480458861635915
