| L(s) = 1 | + 7-s − 2·13-s − 2·19-s + 6·29-s − 8·31-s + 4·37-s + 6·41-s + 2·43-s − 6·47-s + 49-s − 6·53-s − 12·59-s + 8·61-s + 2·67-s − 6·71-s − 2·73-s + 16·79-s + 6·89-s − 2·91-s + 10·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | + 0.377·7-s − 0.554·13-s − 0.458·19-s + 1.11·29-s − 1.43·31-s + 0.657·37-s + 0.937·41-s + 0.304·43-s − 0.875·47-s + 1/7·49-s − 0.824·53-s − 1.56·59-s + 1.02·61-s + 0.244·67-s − 0.712·71-s − 0.234·73-s + 1.80·79-s + 0.635·89-s − 0.209·91-s + 1.01·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−15.74327653804183, −14.89249816008831, −14.58273076716197, −14.21446354934487, −13.37565698650209, −13.01571543266682, −12.28152870928433, −12.01427827567139, −11.14101982895891, −10.86253351018525, −10.22549501378545, −9.509620141388889, −9.139744382481399, −8.399799446217011, −7.821455565177592, −7.391598349508092, −6.585135238874403, −6.124370430855193, −5.316221930912581, −4.778626802806643, −4.180516496276588, −3.408441875414253, −2.620289662568172, −1.962759537151829, −1.073463283450661, 0,
1.073463283450661, 1.962759537151829, 2.620289662568172, 3.408441875414253, 4.180516496276588, 4.778626802806643, 5.316221930912581, 6.124370430855193, 6.585135238874403, 7.391598349508092, 7.821455565177592, 8.399799446217011, 9.139744382481399, 9.509620141388889, 10.22549501378545, 10.86253351018525, 11.14101982895891, 12.01427827567139, 12.28152870928433, 13.01571543266682, 13.37565698650209, 14.21446354934487, 14.58273076716197, 14.89249816008831, 15.74327653804183
