L(s) = 1 | + 11.4·5-s + 29.8·7-s + 66.2·11-s − 39.8·13-s + 107.·17-s + 70.3·19-s − 6.91·23-s + 5.33·25-s − 36.6·29-s + 231.·31-s + 340.·35-s − 36.8·37-s + 429.·41-s − 74.3·43-s − 52.5·47-s + 546.·49-s − 288.·53-s + 756.·55-s − 783.·59-s − 439.·61-s − 454.·65-s − 218.·67-s − 790.·71-s + 1.09e3·73-s + 1.97e3·77-s − 439.·79-s − 50.8·83-s + ⋯ |
L(s) = 1 | + 1.02·5-s + 1.61·7-s + 1.81·11-s − 0.849·13-s + 1.53·17-s + 0.849·19-s − 0.0627·23-s + 0.0426·25-s − 0.234·29-s + 1.34·31-s + 1.64·35-s − 0.163·37-s + 1.63·41-s − 0.263·43-s − 0.163·47-s + 1.59·49-s − 0.746·53-s + 1.85·55-s − 1.72·59-s − 0.921·61-s − 0.867·65-s − 0.398·67-s − 1.32·71-s + 1.76·73-s + 2.92·77-s − 0.626·79-s − 0.0672·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(4.290071831\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.290071831\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 - 11.4T + 125T^{2} \) |
| 7 | \( 1 - 29.8T + 343T^{2} \) |
| 11 | \( 1 - 66.2T + 1.33e3T^{2} \) |
| 13 | \( 1 + 39.8T + 2.19e3T^{2} \) |
| 17 | \( 1 - 107.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 70.3T + 6.85e3T^{2} \) |
| 23 | \( 1 + 6.91T + 1.21e4T^{2} \) |
| 29 | \( 1 + 36.6T + 2.43e4T^{2} \) |
| 31 | \( 1 - 231.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 36.8T + 5.06e4T^{2} \) |
| 41 | \( 1 - 429.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 74.3T + 7.95e4T^{2} \) |
| 47 | \( 1 + 52.5T + 1.03e5T^{2} \) |
| 53 | \( 1 + 288.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 783.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 439.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 218.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 790.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 1.09e3T + 3.89e5T^{2} \) |
| 79 | \( 1 + 439.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 50.8T + 5.71e5T^{2} \) |
| 89 | \( 1 - 719.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.19e3T + 9.12e5T^{2} \) |
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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
−9.210981706177144223936553175816, −8.037734608372937347590407404934, −7.53888082995470534991793904964, −6.45449177217074427172400006373, −5.67554976613937624703579401705, −4.92729727852737634473357027064, −4.09013391643110489201823776612, −2.82290041078521775750871177476, −1.59435355266943973355241035071, −1.18119744194327315900281415741,
1.18119744194327315900281415741, 1.59435355266943973355241035071, 2.82290041078521775750871177476, 4.09013391643110489201823776612, 4.92729727852737634473357027064, 5.67554976613937624703579401705, 6.45449177217074427172400006373, 7.53888082995470534991793904964, 8.037734608372937347590407404934, 9.210981706177144223936553175816