L(s) = 1 | − 3-s − 5-s + 9-s − 11-s + 15-s + 8·17-s − 4·19-s − 4·23-s − 4·25-s − 27-s − 5·29-s + 7·31-s + 33-s + 8·37-s − 4·41-s − 10·43-s − 45-s + 6·47-s − 8·51-s − 53-s + 55-s + 4·57-s + 9·59-s + 2·61-s − 2·67-s + 4·69-s − 6·71-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.447·5-s + 1/3·9-s − 0.301·11-s + 0.258·15-s + 1.94·17-s − 0.917·19-s − 0.834·23-s − 4/5·25-s − 0.192·27-s − 0.928·29-s + 1.25·31-s + 0.174·33-s + 1.31·37-s − 0.624·41-s − 1.52·43-s − 0.149·45-s + 0.875·47-s − 1.12·51-s − 0.137·53-s + 0.134·55-s + 0.529·57-s + 1.17·59-s + 0.256·61-s − 0.244·67-s + 0.481·69-s − 0.712·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4704 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4704 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 8 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 - 7 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 4 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + T + p T^{2} \) |
| 59 | \( 1 - 9 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 9 T + p T^{2} \) |
| 83 | \( 1 + 3 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - T + p T^{2} \) |
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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
−7.946301137869008300609180109916, −7.30839932711627113784225980630, −6.36139311707222520414840065706, −5.77476961643115481345513751822, −5.06174980866984725877358021495, −4.14380299612051020922173438107, −3.50640797261292056094241111843, −2.38782824707232263640154713498, −1.22920135368550025048255947046, 0,
1.22920135368550025048255947046, 2.38782824707232263640154713498, 3.50640797261292056094241111843, 4.14380299612051020922173438107, 5.06174980866984725877358021495, 5.77476961643115481345513751822, 6.36139311707222520414840065706, 7.30839932711627113784225980630, 7.946301137869008300609180109916