| L(s) = 1 | − 5.96·3-s − 7·7-s + 26.5·9-s − 11·11-s − 8.64·13-s − 27.1·17-s + 41.7·21-s + 25·25-s − 104.·27-s − 35.1·31-s + 65.5·33-s − 52.6·37-s + 51.5·39-s − 72.1·41-s − 52.6·43-s − 19.0·47-s + 49·49-s + 161.·51-s − 52.6·53-s + 28.3·59-s − 106.·61-s − 185.·63-s + 99.8·73-s − 149.·75-s + 77·77-s + 157.·79-s + 384.·81-s + ⋯ |
| L(s) = 1 | − 1.98·3-s − 7-s + 2.94·9-s − 11-s − 0.665·13-s − 1.59·17-s + 1.98·21-s + 25-s − 3.87·27-s − 1.13·31-s + 1.98·33-s − 1.42·37-s + 1.32·39-s − 1.75·41-s − 1.22·43-s − 0.405·47-s + 0.999·49-s + 3.17·51-s − 0.993·53-s + 0.479·59-s − 1.74·61-s − 2.94·63-s + 1.36·73-s − 1.98·75-s + 77-s + 1.99·79-s + 4.75·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.1564685226\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1564685226\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + 7T \) |
| 11 | \( 1 + 11T \) |
| good | 3 | \( 1 + 5.96T + 9T^{2} \) |
| 5 | \( 1 - 25T^{2} \) |
| 13 | \( 1 + 8.64T + 169T^{2} \) |
| 17 | \( 1 + 27.1T + 289T^{2} \) |
| 19 | \( 1 - 361T^{2} \) |
| 23 | \( 1 - 529T^{2} \) |
| 29 | \( 1 - 841T^{2} \) |
| 31 | \( 1 + 35.1T + 961T^{2} \) |
| 37 | \( 1 + 52.6T + 1.36e3T^{2} \) |
| 41 | \( 1 + 72.1T + 1.68e3T^{2} \) |
| 43 | \( 1 + 52.6T + 1.84e3T^{2} \) |
| 47 | \( 1 + 19.0T + 2.20e3T^{2} \) |
| 53 | \( 1 + 52.6T + 2.80e3T^{2} \) |
| 59 | \( 1 - 28.3T + 3.48e3T^{2} \) |
| 61 | \( 1 + 106.T + 3.72e3T^{2} \) |
| 67 | \( 1 - 4.48e3T^{2} \) |
| 71 | \( 1 - 5.04e3T^{2} \) |
| 73 | \( 1 - 99.8T + 5.32e3T^{2} \) |
| 79 | \( 1 - 157.T + 6.24e3T^{2} \) |
| 83 | \( 1 - 6.88e3T^{2} \) |
| 89 | \( 1 - 7.92e3T^{2} \) |
| 97 | \( 1 - 9.40e3T^{2} \) |
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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
−9.866203191139990235300281640011, −8.903477721171265608045736171996, −7.48788348035689268113165880102, −6.74548712673059782735823056768, −6.32192413249289585298779440304, −5.12951678516407708420992873905, −4.86801870634779303582481746827, −3.51157028426922835478801756720, −1.93992981350260456977573418676, −0.24680139826550981414275168042,
0.24680139826550981414275168042, 1.93992981350260456977573418676, 3.51157028426922835478801756720, 4.86801870634779303582481746827, 5.12951678516407708420992873905, 6.32192413249289585298779440304, 6.74548712673059782735823056768, 7.48788348035689268113165880102, 8.903477721171265608045736171996, 9.866203191139990235300281640011
