| L(s) = 1 | + 9.27·3-s − 5·5-s − 33.0·7-s + 59.0·9-s + 54.6·11-s + 86.7·13-s − 46.3·15-s + 77.8·17-s − 138.·19-s − 306.·21-s − 23·23-s + 25·25-s + 297.·27-s − 212.·29-s + 147.·31-s + 506.·33-s + 165.·35-s − 185.·37-s + 804.·39-s + 322.·41-s − 3.59·43-s − 295.·45-s + 18.6·47-s + 751.·49-s + 721.·51-s − 6.84·53-s − 273.·55-s + ⋯ |
| L(s) = 1 | + 1.78·3-s − 0.447·5-s − 1.78·7-s + 2.18·9-s + 1.49·11-s + 1.85·13-s − 0.798·15-s + 1.11·17-s − 1.66·19-s − 3.18·21-s − 0.208·23-s + 0.200·25-s + 2.11·27-s − 1.35·29-s + 0.855·31-s + 2.67·33-s + 0.798·35-s − 0.825·37-s + 3.30·39-s + 1.22·41-s − 0.0127·43-s − 0.977·45-s + 0.0578·47-s + 2.19·49-s + 1.98·51-s − 0.0177·53-s − 0.669·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(4.095793780\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.095793780\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + 5T \) |
| 23 | \( 1 + 23T \) |
| good | 3 | \( 1 - 9.27T + 27T^{2} \) |
| 7 | \( 1 + 33.0T + 343T^{2} \) |
| 11 | \( 1 - 54.6T + 1.33e3T^{2} \) |
| 13 | \( 1 - 86.7T + 2.19e3T^{2} \) |
| 17 | \( 1 - 77.8T + 4.91e3T^{2} \) |
| 19 | \( 1 + 138.T + 6.85e3T^{2} \) |
| 29 | \( 1 + 212.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 147.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 185.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 322.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 3.59T + 7.95e4T^{2} \) |
| 47 | \( 1 - 18.6T + 1.03e5T^{2} \) |
| 53 | \( 1 + 6.84T + 1.48e5T^{2} \) |
| 59 | \( 1 + 414.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 258.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 490.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 821.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 11.4T + 3.89e5T^{2} \) |
| 79 | \( 1 + 214.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 793.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 192.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.13e3T + 9.12e5T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.896207901288200056339566109584, −8.351880548410269649354533412591, −7.42717469822902432473324161533, −6.52636387975219503306475145737, −6.10570138778621805945624633489, −4.11779906094298243752173217094, −3.67187041203180370350899814189, −3.27065763385031999742366428636, −2.03808651041576302320578380275, −0.895973185254034978563961330603,
0.895973185254034978563961330603, 2.03808651041576302320578380275, 3.27065763385031999742366428636, 3.67187041203180370350899814189, 4.11779906094298243752173217094, 6.10570138778621805945624633489, 6.52636387975219503306475145737, 7.42717469822902432473324161533, 8.351880548410269649354533412591, 8.896207901288200056339566109584
