L(s) = 1 | + 3-s − 2·4-s − 5-s − 2·9-s + 3·11-s − 2·12-s − 5·13-s − 15-s + 4·16-s − 2·19-s + 2·20-s + 6·23-s + 25-s − 5·27-s − 3·29-s − 4·31-s + 3·33-s + 4·36-s − 2·37-s − 5·39-s − 12·41-s − 10·43-s − 6·44-s + 2·45-s − 9·47-s + 4·48-s + 10·52-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 4-s − 0.447·5-s − 2/3·9-s + 0.904·11-s − 0.577·12-s − 1.38·13-s − 0.258·15-s + 16-s − 0.458·19-s + 0.447·20-s + 1.25·23-s + 1/5·25-s − 0.962·27-s − 0.557·29-s − 0.718·31-s + 0.522·33-s + 2/3·36-s − 0.328·37-s − 0.800·39-s − 1.87·41-s − 1.52·43-s − 0.904·44-s + 0.298·45-s − 1.31·47-s + 0.577·48-s + 1.38·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 70805 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 70805 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4042721547\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4042721547\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| 17 | \( 1 \) |
good | 2 | \( 1 + p T^{2} \) |
| 3 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 + T + p T^{2} \) |
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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
−14.20473318967249, −13.62563655704812, −13.19593792777666, −12.75048760539539, −12.11081942119789, −11.66312287083791, −11.30141119184228, −10.40359787218912, −10.01853952585669, −9.406390252206969, −8.998065122608220, −8.575221282632300, −8.158192064015361, −7.511294819532849, −6.932478450737381, −6.492719509753973, −5.420841516499872, −5.211327431027486, −4.605975129073968, −3.854640259682187, −3.475352201069296, −2.889174743416380, −2.068150886633997, −1.306826257088665, −0.2102226293962212,
0.2102226293962212, 1.306826257088665, 2.068150886633997, 2.889174743416380, 3.475352201069296, 3.854640259682187, 4.605975129073968, 5.211327431027486, 5.420841516499872, 6.492719509753973, 6.932478450737381, 7.511294819532849, 8.158192064015361, 8.575221282632300, 8.998065122608220, 9.406390252206969, 10.01853952585669, 10.40359787218912, 11.30141119184228, 11.66312287083791, 12.11081942119789, 12.75048760539539, 13.19593792777666, 13.62563655704812, 14.20473318967249