| L(s) = 1 | + 3-s − 2·9-s − 11-s − 13-s − 3·17-s + 4·19-s − 2·23-s − 5·27-s − 29-s + 6·31-s − 33-s − 2·37-s − 39-s + 10·41-s + 9·47-s − 3·51-s + 14·53-s + 4·57-s − 6·59-s + 4·61-s − 10·67-s − 2·69-s − 16·71-s + 10·73-s − 11·79-s + 81-s + 4·83-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 2/3·9-s − 0.301·11-s − 0.277·13-s − 0.727·17-s + 0.917·19-s − 0.417·23-s − 0.962·27-s − 0.185·29-s + 1.07·31-s − 0.174·33-s − 0.328·37-s − 0.160·39-s + 1.56·41-s + 1.31·47-s − 0.420·51-s + 1.92·53-s + 0.529·57-s − 0.781·59-s + 0.512·61-s − 1.22·67-s − 0.240·69-s − 1.89·71-s + 1.17·73-s − 1.23·79-s + 1/9·81-s + 0.439·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 - 14 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 + 10 T + p T^{2} \) |
| 71 | \( 1 + 16 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 11 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 + 19 T + p T^{2} \) |
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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
−7.45538144000990900173789206488, −6.75414251940496997637793986908, −5.82507549304985620120574007357, −5.42912598693750670320812127704, −4.41554143045164073098205781986, −3.83046228760266612089739860143, −2.70949310856630956314622300399, −2.54951058000118463522180500026, −1.26407403124555854320478904734, 0,
1.26407403124555854320478904734, 2.54951058000118463522180500026, 2.70949310856630956314622300399, 3.83046228760266612089739860143, 4.41554143045164073098205781986, 5.42912598693750670320812127704, 5.82507549304985620120574007357, 6.75414251940496997637793986908, 7.45538144000990900173789206488
