| L(s) = 1 | + 2.87·3-s + 6.47·5-s − 0.759·9-s − 4.63·11-s + 17.8·13-s + 18.5·15-s + 32.3·19-s + 16.9·25-s − 28.0·27-s + 29·29-s − 61.6·31-s − 13.3·33-s + 51.3·39-s + 40.4·43-s − 4.91·45-s − 92.0·47-s + 49·49-s + 46.9·53-s − 30.0·55-s + 92.7·57-s + 115.·65-s + 48.7·75-s − 137.·79-s − 73.5·81-s + 83.2·87-s − 176.·93-s + 209.·95-s + ⋯ |
| L(s) = 1 | + 0.956·3-s + 1.29·5-s − 0.0843·9-s − 0.421·11-s + 1.37·13-s + 1.23·15-s + 1.70·19-s + 0.678·25-s − 1.03·27-s + 29-s − 1.98·31-s − 0.403·33-s + 1.31·39-s + 0.939·43-s − 0.109·45-s − 1.95·47-s + 0.999·49-s + 0.884·53-s − 0.546·55-s + 1.62·57-s + 1.78·65-s + 0.649·75-s − 1.73·79-s − 0.908·81-s + 0.956·87-s − 1.90·93-s + 2.20·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 464 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 464 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(3.000674077\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.000674077\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 29 | \( 1 - 29T \) |
| good | 3 | \( 1 - 2.87T + 9T^{2} \) |
| 5 | \( 1 - 6.47T + 25T^{2} \) |
| 7 | \( 1 - 49T^{2} \) |
| 11 | \( 1 + 4.63T + 121T^{2} \) |
| 13 | \( 1 - 17.8T + 169T^{2} \) |
| 17 | \( 1 - 289T^{2} \) |
| 19 | \( 1 - 32.3T + 361T^{2} \) |
| 23 | \( 1 - 529T^{2} \) |
| 31 | \( 1 + 61.6T + 961T^{2} \) |
| 37 | \( 1 - 1.36e3T^{2} \) |
| 41 | \( 1 - 1.68e3T^{2} \) |
| 43 | \( 1 - 40.4T + 1.84e3T^{2} \) |
| 47 | \( 1 + 92.0T + 2.20e3T^{2} \) |
| 53 | \( 1 - 46.9T + 2.80e3T^{2} \) |
| 59 | \( 1 - 3.48e3T^{2} \) |
| 61 | \( 1 - 3.72e3T^{2} \) |
| 67 | \( 1 - 4.48e3T^{2} \) |
| 71 | \( 1 - 5.04e3T^{2} \) |
| 73 | \( 1 - 5.32e3T^{2} \) |
| 79 | \( 1 + 137.T + 6.24e3T^{2} \) |
| 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
−10.68245925643516419342836509424, −9.724734407529796819638956113071, −9.090137833528136492385344077292, −8.304229003037192075879527646953, −7.26029705527532511252256795033, −5.99192669045530218870696045275, −5.33745570998349159582527686117, −3.64716186638387479563737615942, −2.67625326763953992587072606669, −1.46508410521703362670743564741,
1.46508410521703362670743564741, 2.67625326763953992587072606669, 3.64716186638387479563737615942, 5.33745570998349159582527686117, 5.99192669045530218870696045275, 7.26029705527532511252256795033, 8.304229003037192075879527646953, 9.090137833528136492385344077292, 9.724734407529796819638956113071, 10.68245925643516419342836509424
