| L(s) = 1 | − 3-s − 5-s + 7-s + 9-s + 2.44·11-s − 0.449·13-s + 15-s − 0.550·17-s + 0.449·19-s − 21-s + 23-s + 25-s − 27-s − 4.34·29-s − 9.89·31-s − 2.44·33-s − 35-s − 5.89·37-s + 0.449·39-s + 0.550·41-s − 2·43-s − 45-s − 3.55·47-s − 6·49-s + 0.550·51-s + 5.44·53-s − 2.44·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 0.377·7-s + 0.333·9-s + 0.738·11-s − 0.124·13-s + 0.258·15-s − 0.133·17-s + 0.103·19-s − 0.218·21-s + 0.208·23-s + 0.200·25-s − 0.192·27-s − 0.807·29-s − 1.77·31-s − 0.426·33-s − 0.169·35-s − 0.969·37-s + 0.0719·39-s + 0.0859·41-s − 0.304·43-s − 0.149·45-s − 0.517·47-s − 0.857·49-s + 0.0770·51-s + 0.748·53-s − 0.330·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| good | 7 | \( 1 - T + 7T^{2} \) |
| 11 | \( 1 - 2.44T + 11T^{2} \) |
| 13 | \( 1 + 0.449T + 13T^{2} \) |
| 17 | \( 1 + 0.550T + 17T^{2} \) |
| 19 | \( 1 - 0.449T + 19T^{2} \) |
| 29 | \( 1 + 4.34T + 29T^{2} \) |
| 31 | \( 1 + 9.89T + 31T^{2} \) |
| 37 | \( 1 + 5.89T + 37T^{2} \) |
| 41 | \( 1 - 0.550T + 41T^{2} \) |
| 43 | \( 1 + 2T + 43T^{2} \) |
| 47 | \( 1 + 3.55T + 47T^{2} \) |
| 53 | \( 1 - 5.44T + 53T^{2} \) |
| 59 | \( 1 - 4.34T + 59T^{2} \) |
| 61 | \( 1 - 15.3T + 61T^{2} \) |
| 67 | \( 1 - 7T + 67T^{2} \) |
| 71 | \( 1 + 10.3T + 71T^{2} \) |
| 73 | \( 1 - 9.34T + 73T^{2} \) |
| 79 | \( 1 - 4T + 79T^{2} \) |
| 83 | \( 1 - 9.24T + 83T^{2} \) |
| 89 | \( 1 + 7.10T + 89T^{2} \) |
| 97 | \( 1 - 12.8T + 97T^{2} \) |
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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
−7.70186646102301411667562043034, −7.02690533959664022719426071644, −6.49851412208641600319588913561, −5.45613047221678302940544463000, −5.05722448009057045034167075535, −3.99294084488089109264212353206, −3.55336477302658470044256243508, −2.19968150372779865790424260110, −1.27416998067059293327661325630, 0,
1.27416998067059293327661325630, 2.19968150372779865790424260110, 3.55336477302658470044256243508, 3.99294084488089109264212353206, 5.05722448009057045034167075535, 5.45613047221678302940544463000, 6.49851412208641600319588913561, 7.02690533959664022719426071644, 7.70186646102301411667562043034
