L(s) = 1 | + 6·11-s − 2·13-s + 4·19-s + 6·23-s − 5·25-s − 6·29-s − 8·31-s + 2·37-s + 12·41-s − 4·43-s + 12·47-s + 6·53-s + 10·61-s + 8·67-s − 6·71-s + 10·73-s − 4·79-s − 12·83-s + 12·89-s + 10·97-s − 12·101-s − 8·103-s + 6·107-s + 14·109-s + 6·113-s + ⋯ |
L(s) = 1 | + 1.80·11-s − 0.554·13-s + 0.917·19-s + 1.25·23-s − 25-s − 1.11·29-s − 1.43·31-s + 0.328·37-s + 1.87·41-s − 0.609·43-s + 1.75·47-s + 0.824·53-s + 1.28·61-s + 0.977·67-s − 0.712·71-s + 1.17·73-s − 0.450·79-s − 1.31·83-s + 1.27·89-s + 1.01·97-s − 1.19·101-s − 0.788·103-s + 0.580·107-s + 1.34·109-s + 0.564·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.911865514\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.911865514\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−9.331187149331773045855656881837, −8.729935221736740713914818138949, −7.44499063873378776873294009662, −7.12758834797060137294534233139, −6.03492627650465326217694729948, −5.32133833421984252130695309604, −4.15927538579926872458234038883, −3.52771007221053761224054171247, −2.20338451459941845252791108478, −1.00219484121657416616236254344,
1.00219484121657416616236254344, 2.20338451459941845252791108478, 3.52771007221053761224054171247, 4.15927538579926872458234038883, 5.32133833421984252130695309604, 6.03492627650465326217694729948, 7.12758834797060137294534233139, 7.44499063873378776873294009662, 8.729935221736740713914818138949, 9.331187149331773045855656881837