| L(s) = 1 | + 2-s + 3-s + 4-s + 1.41·5-s + 6-s + 8-s + 9-s + 1.41·10-s − 2.41·11-s + 12-s + 13-s + 1.41·15-s + 16-s − 3.24·17-s + 18-s + 3·19-s + 1.41·20-s − 2.41·22-s + 5.07·23-s + 24-s − 2.99·25-s + 26-s + 27-s + 5·29-s + 1.41·30-s + 1.17·31-s + 32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.632·5-s + 0.408·6-s + 0.353·8-s + 0.333·9-s + 0.447·10-s − 0.727·11-s + 0.288·12-s + 0.277·13-s + 0.365·15-s + 0.250·16-s − 0.786·17-s + 0.235·18-s + 0.688·19-s + 0.316·20-s − 0.514·22-s + 1.05·23-s + 0.204·24-s − 0.599·25-s + 0.196·26-s + 0.192·27-s + 0.928·29-s + 0.258·30-s + 0.210·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3822 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3822 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.360003585\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.360003585\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 7 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 5 | \( 1 - 1.41T + 5T^{2} \) |
| 11 | \( 1 + 2.41T + 11T^{2} \) |
| 17 | \( 1 + 3.24T + 17T^{2} \) |
| 19 | \( 1 - 3T + 19T^{2} \) |
| 23 | \( 1 - 5.07T + 23T^{2} \) |
| 29 | \( 1 - 5T + 29T^{2} \) |
| 31 | \( 1 - 1.17T + 31T^{2} \) |
| 37 | \( 1 - 9.07T + 37T^{2} \) |
| 41 | \( 1 - 7.41T + 41T^{2} \) |
| 43 | \( 1 + 0.828T + 43T^{2} \) |
| 47 | \( 1 - 9T + 47T^{2} \) |
| 53 | \( 1 + 10.3T + 53T^{2} \) |
| 59 | \( 1 - 10.4T + 59T^{2} \) |
| 61 | \( 1 - 3.24T + 61T^{2} \) |
| 67 | \( 1 + 3.34T + 67T^{2} \) |
| 71 | \( 1 + T + 71T^{2} \) |
| 73 | \( 1 - 13.0T + 73T^{2} \) |
| 79 | \( 1 + 9.31T + 79T^{2} \) |
| 83 | \( 1 - 3.65T + 83T^{2} \) |
| 89 | \( 1 - 2.24T + 89T^{2} \) |
| 97 | \( 1 + 11.8T + 97T^{2} \) |
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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
−8.400251344616668488071078817234, −7.72689131307410011798144234982, −6.95637270481319983302452012980, −6.19415660202671647117484980075, −5.46150629397672533724213061011, −4.69006356954258470328763996880, −3.89842587071899771378089064077, −2.81721772140154688142432070458, −2.37753522456644293595311079088, −1.12365453925280019383108899458,
1.12365453925280019383108899458, 2.37753522456644293595311079088, 2.81721772140154688142432070458, 3.89842587071899771378089064077, 4.69006356954258470328763996880, 5.46150629397672533724213061011, 6.19415660202671647117484980075, 6.95637270481319983302452012980, 7.72689131307410011798144234982, 8.400251344616668488071078817234
