| L(s) = 1 | − 3-s + 9-s − 6·11-s + 2·13-s − 4·17-s + 8·19-s + 8·23-s − 27-s + 8·29-s + 2·31-s + 6·33-s − 2·39-s − 6·41-s + 2·43-s + 2·47-s + 4·51-s − 6·53-s − 8·57-s − 12·59-s + 10·61-s − 2·67-s − 8·69-s + 12·71-s + 10·73-s + 8·79-s + 81-s + 16·83-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/3·9-s − 1.80·11-s + 0.554·13-s − 0.970·17-s + 1.83·19-s + 1.66·23-s − 0.192·27-s + 1.48·29-s + 0.359·31-s + 1.04·33-s − 0.320·39-s − 0.937·41-s + 0.304·43-s + 0.291·47-s + 0.560·51-s − 0.824·53-s − 1.05·57-s − 1.56·59-s + 1.28·61-s − 0.244·67-s − 0.963·69-s + 1.42·71-s + 1.17·73-s + 0.900·79-s + 1/9·81-s + 1.75·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 29400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.655283789\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.655283789\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 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
−15.46633959899658, −14.73859785725648, −13.84725816996390, −13.47862749429398, −13.21972006988009, −12.36225972583557, −12.10525195090679, −11.22573400552624, −10.88989994325262, −10.52922215381395, −9.764278457060653, −9.313877875519477, −8.551785290317632, −7.968460184563260, −7.503855705108848, −6.726029144326841, −6.387605428911441, −5.366346715773429, −5.125070746260099, −4.666135568421191, −3.606228526143485, −2.949900750400077, −2.397058262280194, −1.259663136221077, −0.5636029011231372,
0.5636029011231372, 1.259663136221077, 2.397058262280194, 2.949900750400077, 3.606228526143485, 4.666135568421191, 5.125070746260099, 5.366346715773429, 6.387605428911441, 6.726029144326841, 7.503855705108848, 7.968460184563260, 8.551785290317632, 9.313877875519477, 9.764278457060653, 10.52922215381395, 10.88989994325262, 11.22573400552624, 12.10525195090679, 12.36225972583557, 13.21972006988009, 13.47862749429398, 13.84725816996390, 14.73859785725648, 15.46633959899658
