| L(s) = 1 | + 2·5-s − 2·11-s + 13-s − 19-s − 25-s − 4·29-s − 9·31-s + 3·37-s + 10·41-s − 5·43-s − 6·47-s − 12·53-s − 4·55-s − 12·59-s + 10·61-s + 2·65-s + 5·67-s − 6·71-s − 3·73-s + 79-s + 6·83-s − 16·89-s − 2·95-s − 6·97-s − 2·101-s + 7·103-s − 8·107-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 0.603·11-s + 0.277·13-s − 0.229·19-s − 1/5·25-s − 0.742·29-s − 1.61·31-s + 0.493·37-s + 1.56·41-s − 0.762·43-s − 0.875·47-s − 1.64·53-s − 0.539·55-s − 1.56·59-s + 1.28·61-s + 0.248·65-s + 0.610·67-s − 0.712·71-s − 0.351·73-s + 0.112·79-s + 0.658·83-s − 1.69·89-s − 0.205·95-s − 0.609·97-s − 0.199·101-s + 0.689·103-s − 0.773·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 + 9 T + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 5 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 3 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 16 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−7.64949711306258174096529071054, −6.85436633822462788939980220719, −6.04671932692206981861451685146, −5.60665958130205100053941359967, −4.85143966206839415686987395059, −3.95401218303988875799653774854, −3.08223876131234930334489406795, −2.18667055390494175179886590645, −1.47281845256915450654695175217, 0,
1.47281845256915450654695175217, 2.18667055390494175179886590645, 3.08223876131234930334489406795, 3.95401218303988875799653774854, 4.85143966206839415686987395059, 5.60665958130205100053941359967, 6.04671932692206981861451685146, 6.85436633822462788939980220719, 7.64949711306258174096529071054
