| L(s) = 1 | + 4·7-s − 2·11-s − 13-s + 6·17-s + 4·19-s + 4·23-s − 5·25-s − 10·29-s + 8·31-s − 2·37-s + 4·43-s + 2·47-s + 9·49-s + 2·53-s + 10·59-s + 10·61-s − 8·67-s + 2·71-s − 10·73-s − 8·77-s − 8·79-s + 6·83-s + 12·89-s − 4·91-s − 2·97-s − 2·101-s + 16·103-s + ⋯ |
| L(s) = 1 | + 1.51·7-s − 0.603·11-s − 0.277·13-s + 1.45·17-s + 0.917·19-s + 0.834·23-s − 25-s − 1.85·29-s + 1.43·31-s − 0.328·37-s + 0.609·43-s + 0.291·47-s + 9/7·49-s + 0.274·53-s + 1.30·59-s + 1.28·61-s − 0.977·67-s + 0.237·71-s − 1.17·73-s − 0.911·77-s − 0.900·79-s + 0.658·83-s + 1.27·89-s − 0.419·91-s − 0.203·97-s − 0.199·101-s + 1.57·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.130350495\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.130350495\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−9.239289551943017123934612848385, −8.246518608998310493379441373007, −7.71387236722259124950052322681, −7.17292965380990182887285956413, −5.66602342897317417961265171945, −5.33322403312144877575207388138, −4.39131354020728293864682849714, −3.31414609411084864193635820227, −2.15956775014138729773769000571, −1.06254916932353976871281040816,
1.06254916932353976871281040816, 2.15956775014138729773769000571, 3.31414609411084864193635820227, 4.39131354020728293864682849714, 5.33322403312144877575207388138, 5.66602342897317417961265171945, 7.17292965380990182887285956413, 7.71387236722259124950052322681, 8.246518608998310493379441373007, 9.239289551943017123934612848385
