| L(s) = 1 | + 3-s − 3.70·5-s − 7-s + 9-s − 5.70·11-s + 13-s − 3.70·15-s + 3.70·17-s − 5.70·19-s − 21-s + 1.70·23-s + 8.70·25-s + 27-s − 3.70·29-s − 5.70·33-s + 3.70·35-s + 4.29·37-s + 39-s − 9.40·41-s − 9.10·43-s − 3.70·45-s + 8·47-s + 49-s + 3.70·51-s − 2·53-s + 21.1·55-s − 5.70·57-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.65·5-s − 0.377·7-s + 0.333·9-s − 1.71·11-s + 0.277·13-s − 0.955·15-s + 0.897·17-s − 1.30·19-s − 0.218·21-s + 0.354·23-s + 1.74·25-s + 0.192·27-s − 0.687·29-s − 0.992·33-s + 0.625·35-s + 0.706·37-s + 0.160·39-s − 1.46·41-s − 1.38·43-s − 0.551·45-s + 1.16·47-s + 0.142·49-s + 0.518·51-s − 0.274·53-s + 2.84·55-s − 0.755·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9447762561\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9447762561\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| good | 5 | \( 1 + 3.70T + 5T^{2} \) |
| 11 | \( 1 + 5.70T + 11T^{2} \) |
| 17 | \( 1 - 3.70T + 17T^{2} \) |
| 19 | \( 1 + 5.70T + 19T^{2} \) |
| 23 | \( 1 - 1.70T + 23T^{2} \) |
| 29 | \( 1 + 3.70T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 4.29T + 37T^{2} \) |
| 41 | \( 1 + 9.40T + 41T^{2} \) |
| 43 | \( 1 + 9.10T + 43T^{2} \) |
| 47 | \( 1 - 8T + 47T^{2} \) |
| 53 | \( 1 + 2T + 53T^{2} \) |
| 59 | \( 1 + 10.8T + 59T^{2} \) |
| 61 | \( 1 - 7.70T + 61T^{2} \) |
| 67 | \( 1 - 7.40T + 67T^{2} \) |
| 71 | \( 1 - 8T + 71T^{2} \) |
| 73 | \( 1 + 7.70T + 73T^{2} \) |
| 79 | \( 1 - 3.40T + 79T^{2} \) |
| 83 | \( 1 - 0.596T + 83T^{2} \) |
| 89 | \( 1 - 16.8T + 89T^{2} \) |
| 97 | \( 1 - 16.8T + 97T^{2} \) |
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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
−8.275356686669174909933131698592, −7.73051713583992235681214913288, −7.22905358504426905618177421894, −6.29929345602442334076592727967, −5.21927172234027578866557579167, −4.53404127345482971767876496835, −3.59497070123932052884192639977, −3.16836246562289495840437359481, −2.13717117795071707278366582483, −0.50484284328659524175320708164,
0.50484284328659524175320708164, 2.13717117795071707278366582483, 3.16836246562289495840437359481, 3.59497070123932052884192639977, 4.53404127345482971767876496835, 5.21927172234027578866557579167, 6.29929345602442334076592727967, 7.22905358504426905618177421894, 7.73051713583992235681214913288, 8.275356686669174909933131698592
