| L(s) = 1 | + 3-s + 2.68·5-s − 7-s + 9-s + 11-s − 1.22·13-s + 2.68·15-s − 1.90·17-s + 3·19-s − 21-s + 23-s + 2.22·25-s + 27-s + 6.35·29-s + 9.28·31-s + 33-s − 2.68·35-s + 11.7·37-s − 1.22·39-s − 10.1·41-s − 0.596·43-s + 2.68·45-s + 1.09·47-s + 49-s − 1.90·51-s + 0.222·53-s + 2.68·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.20·5-s − 0.377·7-s + 0.333·9-s + 0.301·11-s − 0.338·13-s + 0.693·15-s − 0.463·17-s + 0.688·19-s − 0.218·21-s + 0.208·23-s + 0.444·25-s + 0.192·27-s + 1.17·29-s + 1.66·31-s + 0.174·33-s − 0.454·35-s + 1.92·37-s − 0.195·39-s − 1.59·41-s − 0.0910·43-s + 0.400·45-s + 0.159·47-s + 0.142·49-s − 0.267·51-s + 0.0305·53-s + 0.362·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1932 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1932 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.722089132\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.722089132\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| good | 5 | \( 1 - 2.68T + 5T^{2} \) |
| 11 | \( 1 - T + 11T^{2} \) |
| 13 | \( 1 + 1.22T + 13T^{2} \) |
| 17 | \( 1 + 1.90T + 17T^{2} \) |
| 19 | \( 1 - 3T + 19T^{2} \) |
| 29 | \( 1 - 6.35T + 29T^{2} \) |
| 31 | \( 1 - 9.28T + 31T^{2} \) |
| 37 | \( 1 - 11.7T + 37T^{2} \) |
| 41 | \( 1 + 10.1T + 41T^{2} \) |
| 43 | \( 1 + 0.596T + 43T^{2} \) |
| 47 | \( 1 - 1.09T + 47T^{2} \) |
| 53 | \( 1 - 0.222T + 53T^{2} \) |
| 59 | \( 1 - 14.9T + 59T^{2} \) |
| 61 | \( 1 + 9.50T + 61T^{2} \) |
| 67 | \( 1 + 1.13T + 67T^{2} \) |
| 71 | \( 1 + 8.41T + 71T^{2} \) |
| 73 | \( 1 + 5.90T + 73T^{2} \) |
| 79 | \( 1 + 2.53T + 79T^{2} \) |
| 83 | \( 1 - 10.3T + 83T^{2} \) |
| 89 | \( 1 + 0.152T + 89T^{2} \) |
| 97 | \( 1 + 0.0904T + 97T^{2} \) |
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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
−9.242815504355724348347113443099, −8.573293280049594289522750149443, −7.67640699699813435469916046174, −6.67500366064856850760689282222, −6.18656764894052608250444320343, −5.15080406218496930464010221437, −4.29296198007361924543133323983, −3.04053465963385953530014928851, −2.37081712774445493943692257538, −1.17019796374813072999146557194,
1.17019796374813072999146557194, 2.37081712774445493943692257538, 3.04053465963385953530014928851, 4.29296198007361924543133323983, 5.15080406218496930464010221437, 6.18656764894052608250444320343, 6.67500366064856850760689282222, 7.67640699699813435469916046174, 8.573293280049594289522750149443, 9.242815504355724348347113443099
