| L(s) = 1 | + 3-s + 5-s − 7-s + 11-s − 13-s + 15-s − 17-s − 21-s + 25-s − 27-s − 29-s + 33-s − 35-s − 39-s + 47-s + 49-s − 51-s + 55-s − 65-s − 2·71-s + 2·73-s + 75-s − 77-s + 79-s − 81-s − 2·83-s − 85-s + ⋯ |
| L(s) = 1 | + 3-s + 5-s − 7-s + 11-s − 13-s + 15-s − 17-s − 21-s + 25-s − 27-s − 29-s + 33-s − 35-s − 39-s + 47-s + 49-s − 51-s + 55-s − 65-s − 2·71-s + 2·73-s + 75-s − 77-s + 79-s − 81-s − 2·83-s − 85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.228345369\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.228345369\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| good | 3 | \( 1 - T + T^{2} \) |
| 11 | \( 1 - T + T^{2} \) |
| 13 | \( 1 + T + T^{2} \) |
| 17 | \( 1 + T + T^{2} \) |
| 19 | \( ( 1 - T )( 1 + T ) \) |
| 23 | \( ( 1 - T )( 1 + T ) \) |
| 29 | \( 1 + T + T^{2} \) |
| 31 | \( ( 1 - T )( 1 + T ) \) |
| 37 | \( ( 1 - T )( 1 + T ) \) |
| 41 | \( ( 1 - T )( 1 + T ) \) |
| 43 | \( ( 1 - T )( 1 + T ) \) |
| 47 | \( 1 - T + T^{2} \) |
| 53 | \( ( 1 - T )( 1 + T ) \) |
| 59 | \( ( 1 - T )( 1 + T ) \) |
| 61 | \( ( 1 - T )( 1 + T ) \) |
| 67 | \( ( 1 - T )( 1 + T ) \) |
| 71 | \( ( 1 + T )^{2} \) |
| 73 | \( ( 1 - T )^{2} \) |
| 79 | \( 1 - T + T^{2} \) |
| 83 | \( ( 1 + T )^{2} \) |
| 89 | \( ( 1 - T )( 1 + T ) \) |
| 97 | \( 1 + T + T^{2} \) |
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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
−10.79525419932982397733887978058, −9.706928676603493420253271919863, −9.316651156830314994879819086668, −8.649450525135616653741343636706, −7.31599171448455487819858083729, −6.51887510112584930764801419479, −5.56657121498221507871856911270, −4.13576206312003803147341858774, −2.97123676879907532825971457021, −2.06224317356630421862300817638,
2.06224317356630421862300817638, 2.97123676879907532825971457021, 4.13576206312003803147341858774, 5.56657121498221507871856911270, 6.51887510112584930764801419479, 7.31599171448455487819858083729, 8.649450525135616653741343636706, 9.316651156830314994879819086668, 9.706928676603493420253271919863, 10.79525419932982397733887978058
