| L(s) = 1 | + 3·3-s − 5.67·5-s − 33.0·7-s + 9·9-s + 34.6·11-s − 82.2·13-s − 17.0·15-s + 97.8·17-s − 55.8·19-s − 99.2·21-s + 130.·23-s − 92.7·25-s + 27·27-s + 147.·29-s − 101.·31-s + 103.·33-s + 187.·35-s + 184.·37-s − 246.·39-s − 237.·41-s + 199.·43-s − 51.0·45-s + 334.·47-s + 752.·49-s + 293.·51-s + 102.·53-s − 196.·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.507·5-s − 1.78·7-s + 0.333·9-s + 0.949·11-s − 1.75·13-s − 0.293·15-s + 1.39·17-s − 0.674·19-s − 1.03·21-s + 1.18·23-s − 0.742·25-s + 0.192·27-s + 0.944·29-s − 0.586·31-s + 0.547·33-s + 0.907·35-s + 0.819·37-s − 1.01·39-s − 0.904·41-s + 0.708·43-s − 0.169·45-s + 1.03·47-s + 2.19·49-s + 0.806·51-s + 0.264·53-s − 0.481·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.566278356\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.566278356\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 3T \) |
| good | 5 | \( 1 + 5.67T + 125T^{2} \) |
| 7 | \( 1 + 33.0T + 343T^{2} \) |
| 11 | \( 1 - 34.6T + 1.33e3T^{2} \) |
| 13 | \( 1 + 82.2T + 2.19e3T^{2} \) |
| 17 | \( 1 - 97.8T + 4.91e3T^{2} \) |
| 19 | \( 1 + 55.8T + 6.85e3T^{2} \) |
| 23 | \( 1 - 130.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 147.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 101.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 184.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 237.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 199.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 334.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 102.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 105.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 717.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 316.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 800.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 301.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 42.8T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.23e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.32e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 505.T + 9.12e5T^{2} \) |
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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
−9.644207955241122414406324764815, −9.385969382338609606066685360523, −8.215614199867146068897771266674, −7.19834467779593724002973766721, −6.71455156282706224704983187238, −5.51333467420216234290244982539, −4.18080781827021538229353021800, −3.36084091836878453283283443734, −2.49382771339511400727310137239, −0.66649278665721321426528852998,
0.66649278665721321426528852998, 2.49382771339511400727310137239, 3.36084091836878453283283443734, 4.18080781827021538229353021800, 5.51333467420216234290244982539, 6.71455156282706224704983187238, 7.19834467779593724002973766721, 8.215614199867146068897771266674, 9.385969382338609606066685360523, 9.644207955241122414406324764815
