L(s) = 1 | + (−0.623 − 0.781i)2-s + (−0.222 + 0.974i)4-s + (0.222 + 0.974i)7-s + (0.900 − 0.433i)8-s + (0.623 − 0.781i)9-s + (1.12 + 1.40i)11-s + (0.623 + 0.781i)13-s + (0.623 − 0.781i)14-s + (−0.900 − 0.433i)16-s + (−0.277 − 1.21i)17-s − 0.999·18-s − 1.24·19-s + (0.400 − 1.75i)22-s + (0.623 − 0.781i)25-s + (0.222 − 0.974i)26-s + ⋯ |
L(s) = 1 | + (−0.623 − 0.781i)2-s + (−0.222 + 0.974i)4-s + (0.222 + 0.974i)7-s + (0.900 − 0.433i)8-s + (0.623 − 0.781i)9-s + (1.12 + 1.40i)11-s + (0.623 + 0.781i)13-s + (0.623 − 0.781i)14-s + (−0.900 − 0.433i)16-s + (−0.277 − 1.21i)17-s − 0.999·18-s − 1.24·19-s + (0.400 − 1.75i)22-s + (0.623 − 0.781i)25-s + (0.222 − 0.974i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2548 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 - 0.127i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2548 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 - 0.127i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9726674775\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9726674775\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.623 + 0.781i)T \) |
| 7 | \( 1 + (-0.222 - 0.974i)T \) |
| 13 | \( 1 + (-0.623 - 0.781i)T \) |
good | 3 | \( 1 + (-0.623 + 0.781i)T^{2} \) |
| 5 | \( 1 + (-0.623 + 0.781i)T^{2} \) |
| 11 | \( 1 + (-1.12 - 1.40i)T + (-0.222 + 0.974i)T^{2} \) |
| 17 | \( 1 + (0.277 + 1.21i)T + (-0.900 + 0.433i)T^{2} \) |
| 19 | \( 1 + 1.24T + T^{2} \) |
| 23 | \( 1 + (0.900 + 0.433i)T^{2} \) |
| 29 | \( 1 + (-0.400 - 1.75i)T + (-0.900 + 0.433i)T^{2} \) |
| 31 | \( 1 + 1.24T + T^{2} \) |
| 37 | \( 1 + (0.900 - 0.433i)T^{2} \) |
| 41 | \( 1 + (-0.623 + 0.781i)T^{2} \) |
| 43 | \( 1 + (-0.623 - 0.781i)T^{2} \) |
| 47 | \( 1 + (-1.12 - 1.40i)T + (-0.222 + 0.974i)T^{2} \) |
| 53 | \( 1 + (0.445 - 1.94i)T + (-0.900 - 0.433i)T^{2} \) |
| 59 | \( 1 + (0.400 + 0.193i)T + (0.623 + 0.781i)T^{2} \) |
| 61 | \( 1 + (0.445 + 1.94i)T + (-0.900 + 0.433i)T^{2} \) |
| 67 | \( 1 - 0.445T + T^{2} \) |
| 71 | \( 1 + (-0.277 + 1.21i)T + (-0.900 - 0.433i)T^{2} \) |
| 73 | \( 1 + (0.222 + 0.974i)T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 + (-0.277 + 0.347i)T + (-0.222 - 0.974i)T^{2} \) |
| 89 | \( 1 + (0.222 + 0.974i)T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−9.056524198829664815906161117751, −8.907936083124650340922509049796, −7.61905587609909426986840796107, −6.84044724117511008089653924367, −6.35674388255373662121523280369, −4.73445706412136860626948757396, −4.33252680528557991642964726779, −3.27091258126448816161472248267, −2.13710632925934534562561376081, −1.41239365191081272110012516078,
0.912869399642643746034070220470, 1.92235456047270173085048507721, 3.74450063105472073156343358942, 4.23208534018140589103589800234, 5.41414831405643461223343703261, 6.15603124082345375193345577441, 6.79329934569825745118713322216, 7.61913556534681115613232340068, 8.426168979198882412168802301938, 8.664310511281954165098725356483