| L(s) = 1 | + 2-s − 3-s + 4-s − 5-s − 6-s + 8-s − 2·9-s − 10-s − 11-s − 12-s + 13-s + 15-s + 16-s − 17-s − 2·18-s − 3·19-s − 20-s − 22-s + 4·23-s − 24-s + 25-s + 26-s + 5·27-s + 3·29-s + 30-s + 31-s + 32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s + 0.353·8-s − 2/3·9-s − 0.316·10-s − 0.301·11-s − 0.288·12-s + 0.277·13-s + 0.258·15-s + 1/4·16-s − 0.242·17-s − 0.471·18-s − 0.688·19-s − 0.223·20-s − 0.213·22-s + 0.834·23-s − 0.204·24-s + 1/5·25-s + 0.196·26-s + 0.962·27-s + 0.557·29-s + 0.182·30-s + 0.179·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 91630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 + 3 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 - T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 + 9 T + p T^{2} \) |
| 61 | \( 1 - 7 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + 9 T + p T^{2} \) |
| 73 | \( 1 - T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 10 T + p T^{2} \) |
| 89 | \( 1 + 15 T + p T^{2} \) |
| 97 | \( 1 - 13 T + p T^{2} \) |
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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
−14.07794246947136, −13.54294121013503, −13.06774455297269, −12.65105347945103, −12.07391994736968, −11.66437120874532, −11.25476612116314, −10.75914958386074, −10.41691662534731, −9.752970267915197, −8.924614046903077, −8.599926818611179, −8.028841090037200, −7.427348734575112, −6.847861170398857, −6.295572068429645, −5.982436962279530, −5.164878430785362, −4.940546573523436, −4.225798619402821, −3.721125519083691, −2.866865452266124, −2.654491706119036, −1.654208421871971, −0.8301610172037857, 0,
0.8301610172037857, 1.654208421871971, 2.654491706119036, 2.866865452266124, 3.721125519083691, 4.225798619402821, 4.940546573523436, 5.164878430785362, 5.982436962279530, 6.295572068429645, 6.847861170398857, 7.427348734575112, 8.028841090037200, 8.599926818611179, 8.924614046903077, 9.752970267915197, 10.41691662534731, 10.75914958386074, 11.25476612116314, 11.66437120874532, 12.07391994736968, 12.65105347945103, 13.06774455297269, 13.54294121013503, 14.07794246947136
