| L(s) = 1 | − 2-s + 3-s + 4-s − 6-s − 8-s − 11-s + 12-s + 16-s + 2·17-s + 22-s − 24-s + 25-s − 27-s − 32-s − 33-s − 2·34-s + 41-s + 2·43-s − 44-s + 48-s + 49-s − 50-s + 2·51-s + 54-s + 59-s + 64-s + 66-s + ⋯ |
| L(s) = 1 | − 2-s + 3-s + 4-s − 6-s − 8-s − 11-s + 12-s + 16-s + 2·17-s + 22-s − 24-s + 25-s − 27-s − 32-s − 33-s − 2·34-s + 41-s + 2·43-s − 44-s + 48-s + 49-s − 50-s + 2·51-s + 54-s + 59-s + 64-s + 66-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.101410573\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.101410573\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 - T + T^{2} \) |
| 5 | \( ( 1 - T )( 1 + T ) \) |
| 7 | \( ( 1 - T )( 1 + T ) \) |
| 11 | \( 1 + T + T^{2} \) |
| 13 | \( ( 1 - T )( 1 + T ) \) |
| 17 | \( ( 1 - T )^{2} \) |
| 23 | \( ( 1 - T )( 1 + T ) \) |
| 29 | \( ( 1 - T )( 1 + T ) \) |
| 31 | \( ( 1 - T )( 1 + T ) \) |
| 37 | \( ( 1 - T )( 1 + T ) \) |
| 41 | \( 1 - T + T^{2} \) |
| 43 | \( ( 1 - T )^{2} \) |
| 47 | \( ( 1 - T )( 1 + T ) \) |
| 53 | \( ( 1 - T )( 1 + T ) \) |
| 59 | \( 1 - T + T^{2} \) |
| 61 | \( ( 1 - T )( 1 + T ) \) |
| 67 | \( 1 - T + T^{2} \) |
| 71 | \( ( 1 - T )( 1 + T ) \) |
| 73 | \( 1 + T + T^{2} \) |
| 79 | \( ( 1 - T )( 1 + T ) \) |
| 83 | \( 1 + T + T^{2} \) |
| 89 | \( ( 1 + T )^{2} \) |
| 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
−8.881312106132044865701369671213, −8.224392775461817345443693298339, −7.66391606499565560396663286736, −7.16361705142421129201980894646, −5.89612440855770135114792140675, −5.37485341119105290604623903952, −3.87824894740151656026907328436, −2.91675624112143575955609089100, −2.47571062173348506407521407858, −1.09762226500840732611717740449,
1.09762226500840732611717740449, 2.47571062173348506407521407858, 2.91675624112143575955609089100, 3.87824894740151656026907328436, 5.37485341119105290604623903952, 5.89612440855770135114792140675, 7.16361705142421129201980894646, 7.66391606499565560396663286736, 8.224392775461817345443693298339, 8.881312106132044865701369671213
