L(s) = 1 | − 3-s − 2·5-s − 7-s − 2·9-s − 11-s − 3·13-s + 2·15-s + 4·17-s + 21-s + 7·23-s − 25-s + 5·27-s − 4·29-s − 2·31-s + 33-s + 2·35-s + 8·37-s + 3·39-s − 10·41-s + 11·43-s + 4·45-s + 47-s − 6·49-s − 4·51-s + 2·55-s + 4·59-s + 5·61-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.894·5-s − 0.377·7-s − 2/3·9-s − 0.301·11-s − 0.832·13-s + 0.516·15-s + 0.970·17-s + 0.218·21-s + 1.45·23-s − 1/5·25-s + 0.962·27-s − 0.742·29-s − 0.359·31-s + 0.174·33-s + 0.338·35-s + 1.31·37-s + 0.480·39-s − 1.56·41-s + 1.67·43-s + 0.596·45-s + 0.145·47-s − 6/7·49-s − 0.560·51-s + 0.269·55-s + 0.520·59-s + 0.640·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 503 | \( 1 - T \) |
good | 3 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 7 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 11 T + p T^{2} \) |
| 47 | \( 1 - T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 - 3 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 - 11 T + p T^{2} \) |
| 89 | \( 1 + 16 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−7.47606786392507957224830314900, −6.88357738459420844800379723355, −6.02860053274972548594954516555, −5.34441023935546280532519572452, −4.83534183169675428741340649804, −3.83755713191031738543105483783, −3.16745546266349881380387091646, −2.38958792919646841280749174956, −0.918589251471694795112986634251, 0,
0.918589251471694795112986634251, 2.38958792919646841280749174956, 3.16745546266349881380387091646, 3.83755713191031738543105483783, 4.83534183169675428741340649804, 5.34441023935546280532519572452, 6.02860053274972548594954516555, 6.88357738459420844800379723355, 7.47606786392507957224830314900