| L(s) = 1 | + 3-s − 5-s − 7-s + 9-s − 6·11-s − 15-s + 17-s − 6·19-s − 21-s − 4·23-s + 25-s + 27-s + 4·29-s + 8·31-s − 6·33-s + 35-s − 4·37-s − 2·41-s + 12·43-s − 45-s + 8·47-s + 49-s + 51-s + 6·53-s + 6·55-s − 6·57-s − 14·59-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s − 0.377·7-s + 1/3·9-s − 1.80·11-s − 0.258·15-s + 0.242·17-s − 1.37·19-s − 0.218·21-s − 0.834·23-s + 1/5·25-s + 0.192·27-s + 0.742·29-s + 1.43·31-s − 1.04·33-s + 0.169·35-s − 0.657·37-s − 0.312·41-s + 1.82·43-s − 0.149·45-s + 1.16·47-s + 1/7·49-s + 0.140·51-s + 0.824·53-s + 0.809·55-s − 0.794·57-s − 1.82·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 114240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 114240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8885242440\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8885242440\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| good | 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 14 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
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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
−13.67787968278734, −13.21971647221156, −12.58633844314566, −12.23585996571269, −11.94672361542881, −10.90514005490328, −10.62689269643271, −10.33685383485620, −9.757429535669296, −9.126932056465584, −8.574974450771609, −8.106608728645077, −7.851597545607671, −7.208380185562023, −6.722247749115988, −5.958354823615631, −5.649556283995703, −4.792492913372097, −4.336007591492851, −3.910748934752535, −2.933353106168344, −2.751451461681811, −2.169684493831807, −1.243851823220915, −0.2805714255996138,
0.2805714255996138, 1.243851823220915, 2.169684493831807, 2.751451461681811, 2.933353106168344, 3.910748934752535, 4.336007591492851, 4.792492913372097, 5.649556283995703, 5.958354823615631, 6.722247749115988, 7.208380185562023, 7.851597545607671, 8.106608728645077, 8.574974450771609, 9.126932056465584, 9.757429535669296, 10.33685383485620, 10.62689269643271, 10.90514005490328, 11.94672361542881, 12.23585996571269, 12.58633844314566, 13.21971647221156, 13.67787968278734
