L(s) = 1 | − 3-s + 2·5-s + 9-s + 2·11-s + 3·13-s − 2·15-s + 8·17-s − 19-s − 8·23-s − 25-s − 27-s − 4·29-s − 3·31-s − 2·33-s + 37-s − 3·39-s + 6·41-s + 11·43-s + 2·45-s − 6·47-s − 8·51-s + 12·53-s + 4·55-s + 57-s + 4·59-s + 6·61-s + 6·65-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.894·5-s + 1/3·9-s + 0.603·11-s + 0.832·13-s − 0.516·15-s + 1.94·17-s − 0.229·19-s − 1.66·23-s − 1/5·25-s − 0.192·27-s − 0.742·29-s − 0.538·31-s − 0.348·33-s + 0.164·37-s − 0.480·39-s + 0.937·41-s + 1.67·43-s + 0.298·45-s − 0.875·47-s − 1.12·51-s + 1.64·53-s + 0.539·55-s + 0.132·57-s + 0.520·59-s + 0.768·61-s + 0.744·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9408 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9408 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.384153980\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.384153980\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 \) |
good | 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 - 8 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 + 3 T + p T^{2} \) |
| 37 | \( 1 - T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 11 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 13 T + p T^{2} \) |
| 71 | \( 1 - 10 T + p T^{2} \) |
| 73 | \( 1 + 11 T + p T^{2} \) |
| 79 | \( 1 - 3 T + p T^{2} \) |
| 83 | \( 1 - 2 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−7.71519968134024182611424889035, −6.90888822361407424677246526941, −6.05625089432050319985415467128, −5.79487726248577525993285155430, −5.25492808350590913990135380677, −3.97304228690350389627883632147, −3.74825195432256222169931740464, −2.43911737360755865843895106307, −1.62808334583662351932437715652, −0.809985284739777375050527012734,
0.809985284739777375050527012734, 1.62808334583662351932437715652, 2.43911737360755865843895106307, 3.74825195432256222169931740464, 3.97304228690350389627883632147, 5.25492808350590913990135380677, 5.79487726248577525993285155430, 6.05625089432050319985415467128, 6.90888822361407424677246526941, 7.71519968134024182611424889035