L(s) = 1 | − 3·3-s − 3·5-s − 2·7-s + 6·9-s − 11-s + 9·15-s − 6·17-s + 4·19-s + 6·21-s + 23-s + 4·25-s − 9·27-s − 8·29-s − 7·31-s + 3·33-s + 6·35-s − 37-s + 4·41-s + 6·43-s − 18·45-s − 8·47-s − 3·49-s + 18·51-s + 2·53-s + 3·55-s − 12·57-s − 59-s + ⋯ |
L(s) = 1 | − 1.73·3-s − 1.34·5-s − 0.755·7-s + 2·9-s − 0.301·11-s + 2.32·15-s − 1.45·17-s + 0.917·19-s + 1.30·21-s + 0.208·23-s + 4/5·25-s − 1.73·27-s − 1.48·29-s − 1.25·31-s + 0.522·33-s + 1.01·35-s − 0.164·37-s + 0.624·41-s + 0.914·43-s − 2.68·45-s − 1.16·47-s − 3/7·49-s + 2.52·51-s + 0.274·53-s + 0.404·55-s − 1.58·57-s − 0.130·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 88 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 88 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 + T \) |
good | 3 | \( 1 + p T + p T^{2} \) |
| 5 | \( 1 + 3 T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 + 7 T + p T^{2} \) |
| 37 | \( 1 + T + p T^{2} \) |
| 41 | \( 1 - 4 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 + 5 T + p T^{2} \) |
| 71 | \( 1 - 3 T + p T^{2} \) |
| 73 | \( 1 - 16 T + p T^{2} \) |
| 79 | \( 1 - 2 T + p T^{2} \) |
| 83 | \( 1 + 2 T + p T^{2} \) |
| 89 | \( 1 - 15 T + p T^{2} \) |
| 97 | \( 1 + 7 T + p T^{2} \) |
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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
−13.16442893918528910310079055859, −12.38704213422326524860027303076, −11.36466946676257259431485654443, −10.89204680724752420498113582025, −9.401670233685537405973889044671, −7.57356519540538183281509096087, −6.61017530637467554226176574176, −5.26718062023912132586056594917, −3.92131200412791242288608029354, 0,
3.92131200412791242288608029354, 5.26718062023912132586056594917, 6.61017530637467554226176574176, 7.57356519540538183281509096087, 9.401670233685537405973889044671, 10.89204680724752420498113582025, 11.36466946676257259431485654443, 12.38704213422326524860027303076, 13.16442893918528910310079055859