| L(s) = 1 | − 2.67e3·2-s + 4.78e6·3-s − 5.29e8·4-s + 1.94e10·5-s − 1.27e10·6-s − 6.78e11·7-s + 2.85e12·8-s + 2.28e13·9-s − 5.20e13·10-s − 2.34e15·11-s − 2.53e15·12-s − 1.06e16·13-s + 1.81e15·14-s + 9.31e16·15-s + 2.76e17·16-s − 1.80e17·17-s − 6.11e16·18-s + 3.63e18·19-s − 1.03e19·20-s − 3.24e18·21-s + 6.26e18·22-s + 1.42e19·23-s + 1.36e19·24-s + 1.92e20·25-s + 2.85e19·26-s + 1.09e20·27-s + 3.59e20·28-s + ⋯ |
| L(s) = 1 | − 0.115·2-s + 0.577·3-s − 0.986·4-s + 1.42·5-s − 0.0665·6-s − 0.377·7-s + 0.229·8-s + 0.333·9-s − 0.164·10-s − 1.86·11-s − 0.569·12-s − 0.751·13-s + 0.0435·14-s + 0.823·15-s + 0.960·16-s − 0.259·17-s − 0.0384·18-s + 1.04·19-s − 1.40·20-s − 0.218·21-s + 0.214·22-s + 0.256·23-s + 0.132·24-s + 1.03·25-s + 0.0866·26-s + 0.192·27-s + 0.372·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(30-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+29/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(15)\) |
\(\approx\) |
\(2.055491021\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.055491021\) |
| \(L(\frac{31}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 4.78e6T \) |
| 7 | \( 1 + 6.78e11T \) |
| good | 2 | \( 1 + 2.67e3T + 5.36e8T^{2} \) |
| 5 | \( 1 - 1.94e10T + 1.86e20T^{2} \) |
| 11 | \( 1 + 2.34e15T + 1.58e30T^{2} \) |
| 13 | \( 1 + 1.06e16T + 2.01e32T^{2} \) |
| 17 | \( 1 + 1.80e17T + 4.81e35T^{2} \) |
| 19 | \( 1 - 3.63e18T + 1.21e37T^{2} \) |
| 23 | \( 1 - 1.42e19T + 3.09e39T^{2} \) |
| 29 | \( 1 - 1.38e21T + 2.56e42T^{2} \) |
| 31 | \( 1 + 1.39e21T + 1.77e43T^{2} \) |
| 37 | \( 1 - 1.35e22T + 3.00e45T^{2} \) |
| 41 | \( 1 + 4.23e23T + 5.89e46T^{2} \) |
| 43 | \( 1 - 3.71e23T + 2.34e47T^{2} \) |
| 47 | \( 1 - 2.04e24T + 3.09e48T^{2} \) |
| 53 | \( 1 - 1.15e25T + 1.00e50T^{2} \) |
| 59 | \( 1 + 2.23e25T + 2.26e51T^{2} \) |
| 61 | \( 1 + 4.97e25T + 5.95e51T^{2} \) |
| 67 | \( 1 - 5.48e26T + 9.04e52T^{2} \) |
| 71 | \( 1 - 1.06e27T + 4.85e53T^{2} \) |
| 73 | \( 1 + 1.06e27T + 1.08e54T^{2} \) |
| 79 | \( 1 + 5.67e27T + 1.07e55T^{2} \) |
| 83 | \( 1 + 5.28e27T + 4.50e55T^{2} \) |
| 89 | \( 1 - 2.36e28T + 3.40e56T^{2} \) |
| 97 | \( 1 - 2.86e28T + 4.13e57T^{2} \) |
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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
−12.68115938954535052419345979427, −10.29328048542284842279148345609, −9.774628315254027279230915806239, −8.678135866656270294472598056844, −7.39637733052465812169567729947, −5.64319323296118817252737842331, −4.84781427445195659745427917765, −3.08270668523267058499863535515, −2.14300836172957405047834922260, −0.66112993287231012313637232803,
0.66112993287231012313637232803, 2.14300836172957405047834922260, 3.08270668523267058499863535515, 4.84781427445195659745427917765, 5.64319323296118817252737842331, 7.39637733052465812169567729947, 8.678135866656270294472598056844, 9.774628315254027279230915806239, 10.29328048542284842279148345609, 12.68115938954535052419345979427
