| L(s) = 1 | + 2-s + 4-s + 5-s − 7-s + 8-s + 10-s − 14-s + 16-s + 20-s − 23-s + 25-s − 28-s − 29-s + 32-s − 35-s + 40-s − 41-s + 2·43-s − 46-s − 47-s + 50-s − 56-s − 58-s − 61-s + 64-s − 67-s − 70-s + ⋯ |
| L(s) = 1 | + 2-s + 4-s + 5-s − 7-s + 8-s + 10-s − 14-s + 16-s + 20-s − 23-s + 25-s − 28-s − 29-s + 32-s − 35-s + 40-s − 41-s + 2·43-s − 46-s − 47-s + 50-s − 56-s − 58-s − 61-s + 64-s − 67-s − 70-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(2.232959799\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.232959799\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| good | 7 | \( 1 + T + T^{2} \) |
| 11 | \( ( 1 - T )( 1 + T ) \) |
| 13 | \( ( 1 - T )( 1 + T ) \) |
| 17 | \( ( 1 - T )( 1 + T ) \) |
| 19 | \( ( 1 - T )( 1 + T ) \) |
| 23 | \( 1 + T + T^{2} \) |
| 29 | \( 1 + T + T^{2} \) |
| 31 | \( ( 1 - T )( 1 + T ) \) |
| 37 | \( ( 1 - T )( 1 + T ) \) |
| 41 | \( 1 + T + T^{2} \) |
| 43 | \( ( 1 - T )^{2} \) |
| 47 | \( 1 + T + T^{2} \) |
| 53 | \( ( 1 - T )( 1 + T ) \) |
| 59 | \( ( 1 - T )( 1 + T ) \) |
| 61 | \( 1 + T + T^{2} \) |
| 67 | \( 1 + T + T^{2} \) |
| 71 | \( ( 1 - T )( 1 + T ) \) |
| 73 | \( ( 1 - T )( 1 + T ) \) |
| 79 | \( ( 1 - T )( 1 + T ) \) |
| 83 | \( 1 + T + T^{2} \) |
| 89 | \( 1 + T + T^{2} \) |
| 97 | \( ( 1 - T )( 1 + T ) \) |
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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
−9.770066527798407030641800202621, −8.948784857459134369409836100195, −7.75213059564737046044151089250, −6.89271979896496467607141958348, −6.12670422834020255834456518706, −5.68049522650013154759787686723, −4.63533792276988180584356757998, −3.60722555458245745861159199434, −2.73180554100524686081832634562, −1.72472024463858295361250271996,
1.72472024463858295361250271996, 2.73180554100524686081832634562, 3.60722555458245745861159199434, 4.63533792276988180584356757998, 5.68049522650013154759787686723, 6.12670422834020255834456518706, 6.89271979896496467607141958348, 7.75213059564737046044151089250, 8.948784857459134369409836100195, 9.770066527798407030641800202621
