| L(s) = 1 | − 3·3-s − 10.8·5-s − 32.1·7-s + 9·9-s − 30.2·11-s + 13·13-s + 32.4·15-s + 34·17-s + 41.8·19-s + 96.5·21-s + 45.5·23-s − 8.28·25-s − 27·27-s + 2.40·29-s + 73.7·31-s + 90.6·33-s + 347.·35-s + 401.·37-s − 39·39-s − 353.·41-s − 329.·43-s − 97.2·45-s + 45.1·47-s + 693.·49-s − 102·51-s + 449.·53-s + 326.·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.966·5-s − 1.73·7-s + 0.333·9-s − 0.828·11-s + 0.277·13-s + 0.557·15-s + 0.485·17-s + 0.505·19-s + 1.00·21-s + 0.412·23-s − 0.0662·25-s − 0.192·27-s + 0.0154·29-s + 0.427·31-s + 0.478·33-s + 1.67·35-s + 1.78·37-s − 0.160·39-s − 1.34·41-s − 1.16·43-s − 0.322·45-s + 0.140·47-s + 2.02·49-s − 0.280·51-s + 1.16·53-s + 0.800·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 312 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.6548537293\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6548537293\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 3T \) |
| 13 | \( 1 - 13T \) |
| good | 5 | \( 1 + 10.8T + 125T^{2} \) |
| 7 | \( 1 + 32.1T + 343T^{2} \) |
| 11 | \( 1 + 30.2T + 1.33e3T^{2} \) |
| 17 | \( 1 - 34T + 4.91e3T^{2} \) |
| 19 | \( 1 - 41.8T + 6.85e3T^{2} \) |
| 23 | \( 1 - 45.5T + 1.21e4T^{2} \) |
| 29 | \( 1 - 2.40T + 2.43e4T^{2} \) |
| 31 | \( 1 - 73.7T + 2.97e4T^{2} \) |
| 37 | \( 1 - 401.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 353.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 329.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 45.1T + 1.03e5T^{2} \) |
| 53 | \( 1 - 449.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 351.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 872.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 177.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 32.9T + 3.57e5T^{2} \) |
| 73 | \( 1 - 777.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 350.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 421.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.36e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 468.T + 9.12e5T^{2} \) |
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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
−11.35473228281097156889632899754, −10.25180959530670060989192422382, −9.615670107138524968739507667835, −8.287754698945801424664056837319, −7.27583271307434187580091015543, −6.38855428497556751475640986077, −5.32044198889573947734068103543, −3.90986639508482489353299048445, −2.95184291260975711182298362384, −0.55106253644359565211371329101,
0.55106253644359565211371329101, 2.95184291260975711182298362384, 3.90986639508482489353299048445, 5.32044198889573947734068103543, 6.38855428497556751475640986077, 7.27583271307434187580091015543, 8.287754698945801424664056837319, 9.615670107138524968739507667835, 10.25180959530670060989192422382, 11.35473228281097156889632899754
