| L(s) = 1 | − 3-s − 3·5-s + 7-s + 9-s − 4·13-s + 3·15-s − 17-s − 5·19-s − 21-s + 4·25-s − 27-s + 6·29-s + 7·31-s − 3·35-s − 37-s + 4·39-s − 11·43-s − 3·45-s + 6·47-s − 6·49-s + 51-s + 6·53-s + 5·57-s − 15·59-s − 61-s + 63-s + 12·65-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.34·5-s + 0.377·7-s + 1/3·9-s − 1.10·13-s + 0.774·15-s − 0.242·17-s − 1.14·19-s − 0.218·21-s + 4/5·25-s − 0.192·27-s + 1.11·29-s + 1.25·31-s − 0.507·35-s − 0.164·37-s + 0.640·39-s − 1.67·43-s − 0.447·45-s + 0.875·47-s − 6/7·49-s + 0.140·51-s + 0.824·53-s + 0.662·57-s − 1.95·59-s − 0.128·61-s + 0.125·63-s + 1.48·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 98736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 11 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 5 | \( 1 + 3 T + p T^{2} \) |
| 7 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 7 T + p T^{2} \) |
| 37 | \( 1 + T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 11 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 15 T + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - 13 T + p T^{2} \) |
| 83 | \( 1 + 9 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 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
−13.97254439367178, −13.58303528722887, −12.77664831913333, −12.44742717269365, −11.92565822308229, −11.74906390500813, −11.08632995700526, −10.73391859907599, −10.05669884273605, −9.819463592797346, −8.787013426042783, −8.565840522818039, −7.885903048712464, −7.602536466480202, −6.899864222075848, −6.533103307635415, −5.948557732212142, −5.040839155552825, −4.701050690059446, −4.361041185317281, −3.682833204037513, −2.978303328887996, −2.340654145110302, −1.540680379556788, −0.6253883288929129, 0,
0.6253883288929129, 1.540680379556788, 2.340654145110302, 2.978303328887996, 3.682833204037513, 4.361041185317281, 4.701050690059446, 5.040839155552825, 5.948557732212142, 6.533103307635415, 6.899864222075848, 7.602536466480202, 7.885903048712464, 8.565840522818039, 8.787013426042783, 9.819463592797346, 10.05669884273605, 10.73391859907599, 11.08632995700526, 11.74906390500813, 11.92565822308229, 12.44742717269365, 12.77664831913333, 13.58303528722887, 13.97254439367178
