| L(s) = 1 | − 3-s − 4·7-s + 9-s − 13-s − 5·19-s + 4·21-s − 27-s − 3·29-s − 4·31-s + 7·37-s + 39-s + 3·41-s − 2·43-s + 9·47-s + 9·49-s − 9·53-s + 5·57-s − 6·59-s − 8·61-s − 4·63-s − 5·67-s − 3·71-s − 4·73-s + 11·79-s + 81-s + 6·83-s + 3·87-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.51·7-s + 1/3·9-s − 0.277·13-s − 1.14·19-s + 0.872·21-s − 0.192·27-s − 0.557·29-s − 0.718·31-s + 1.15·37-s + 0.160·39-s + 0.468·41-s − 0.304·43-s + 1.31·47-s + 9/7·49-s − 1.23·53-s + 0.662·57-s − 0.781·59-s − 1.02·61-s − 0.503·63-s − 0.610·67-s − 0.356·71-s − 0.468·73-s + 1.23·79-s + 1/9·81-s + 0.658·83-s + 0.321·87-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 62400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 62400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2973744975\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2973744975\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 7 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 5 T + p T^{2} \) |
| 71 | \( 1 + 3 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - 11 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−14.30866547002786, −13.47564212710571, −13.28708371555753, −12.65340171690906, −12.35774061457731, −11.90983338085365, −10.99901612472245, −10.83116355696825, −10.23472510975716, −9.613903610652446, −9.258969175534280, −8.821015817022692, −7.841257566540626, −7.550076645720394, −6.721321916614084, −6.433040900040513, −5.916331494817457, −5.402859842342877, −4.560994133938580, −4.096131074315853, −3.439136463410053, −2.781437205580997, −2.141541679393143, −1.214121109407429, −0.2001157127338497,
0.2001157127338497, 1.214121109407429, 2.141541679393143, 2.781437205580997, 3.439136463410053, 4.096131074315853, 4.560994133938580, 5.402859842342877, 5.916331494817457, 6.433040900040513, 6.721321916614084, 7.550076645720394, 7.841257566540626, 8.821015817022692, 9.258969175534280, 9.613903610652446, 10.23472510975716, 10.83116355696825, 10.99901612472245, 11.90983338085365, 12.35774061457731, 12.65340171690906, 13.28708371555753, 13.47564212710571, 14.30866547002786
