| L(s) = 1 | − 5·3-s + 5·5-s − 7·7-s − 2·9-s + 39·11-s + 19·13-s − 25·15-s − 37·17-s + 18·19-s + 35·21-s − 90·23-s + 25·25-s + 145·27-s − 99·29-s − 32·31-s − 195·33-s − 35·35-s − 46·37-s − 95·39-s − 248·41-s − 178·43-s − 10·45-s + 429·47-s + 49·49-s + 185·51-s + 652·53-s + 195·55-s + ⋯ |
| L(s) = 1 | − 0.962·3-s + 0.447·5-s − 0.377·7-s − 0.0740·9-s + 1.06·11-s + 0.405·13-s − 0.430·15-s − 0.527·17-s + 0.217·19-s + 0.363·21-s − 0.815·23-s + 1/5·25-s + 1.03·27-s − 0.633·29-s − 0.185·31-s − 1.02·33-s − 0.169·35-s − 0.204·37-s − 0.390·39-s − 0.944·41-s − 0.631·43-s − 0.0331·45-s + 1.33·47-s + 1/7·49-s + 0.507·51-s + 1.68·53-s + 0.478·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - p T \) |
| 7 | \( 1 + p T \) |
| good | 3 | \( 1 + 5 T + p^{3} T^{2} \) |
| 11 | \( 1 - 39 T + p^{3} T^{2} \) |
| 13 | \( 1 - 19 T + p^{3} T^{2} \) |
| 17 | \( 1 + 37 T + p^{3} T^{2} \) |
| 19 | \( 1 - 18 T + p^{3} T^{2} \) |
| 23 | \( 1 + 90 T + p^{3} T^{2} \) |
| 29 | \( 1 + 99 T + p^{3} T^{2} \) |
| 31 | \( 1 + 32 T + p^{3} T^{2} \) |
| 37 | \( 1 + 46 T + p^{3} T^{2} \) |
| 41 | \( 1 + 248 T + p^{3} T^{2} \) |
| 43 | \( 1 + 178 T + p^{3} T^{2} \) |
| 47 | \( 1 - 429 T + p^{3} T^{2} \) |
| 53 | \( 1 - 652 T + p^{3} T^{2} \) |
| 59 | \( 1 + 40 T + p^{3} T^{2} \) |
| 61 | \( 1 - 36 T + p^{3} T^{2} \) |
| 67 | \( 1 - 348 T + p^{3} T^{2} \) |
| 71 | \( 1 - 72 T + p^{3} T^{2} \) |
| 73 | \( 1 + 1190 T + p^{3} T^{2} \) |
| 79 | \( 1 - 699 T + p^{3} T^{2} \) |
| 83 | \( 1 - 116 T + p^{3} T^{2} \) |
| 89 | \( 1 + 704 T + p^{3} T^{2} \) |
| 97 | \( 1 - 223 T + p^{3} T^{2} \) |
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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
−8.511117673126547659567430732463, −7.25189154064827213201002210159, −6.56568326448918545564212782227, −5.95664899297514281732710219791, −5.34245164158620262955664172314, −4.28864500751552925780246336028, −3.44568049920182101174699252705, −2.19027415317021058192374152145, −1.09634587884136787872410276079, 0,
1.09634587884136787872410276079, 2.19027415317021058192374152145, 3.44568049920182101174699252705, 4.28864500751552925780246336028, 5.34245164158620262955664172314, 5.95664899297514281732710219791, 6.56568326448918545564212782227, 7.25189154064827213201002210159, 8.511117673126547659567430732463
