| L(s) = 1 | + 6.55e4·2-s + 9.85e7·3-s + 4.29e9·4-s + 3.74e11·5-s + 6.45e12·6-s − 1.82e13·7-s + 2.81e14·8-s + 4.14e15·9-s + 2.45e16·10-s − 2.87e17·11-s + 4.23e17·12-s + 3.77e18·13-s − 1.19e18·14-s + 3.68e19·15-s + 1.84e19·16-s + 2.29e20·17-s + 2.71e20·18-s − 1.83e20·19-s + 1.60e21·20-s − 1.79e21·21-s − 1.88e22·22-s − 4.16e22·23-s + 2.77e22·24-s + 2.37e22·25-s + 2.47e23·26-s − 1.38e23·27-s − 7.81e22·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.32·3-s + 0.5·4-s + 1.09·5-s + 0.934·6-s − 0.207·7-s + 0.353·8-s + 0.746·9-s + 0.775·10-s − 1.88·11-s + 0.660·12-s + 1.57·13-s − 0.146·14-s + 1.44·15-s + 0.250·16-s + 1.14·17-s + 0.527·18-s − 0.145·19-s + 0.548·20-s − 0.273·21-s − 1.33·22-s − 1.41·23-s + 0.467·24-s + 0.203·25-s + 1.11·26-s − 0.335·27-s − 0.103·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(34-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2 ^{s/2} \, \Gamma_{\C}(s+33/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(17)\) |
\(\approx\) |
\(4.545928050\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.545928050\) |
| \(L(\frac{35}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 6.55e4T \) |
| good | 3 | \( 1 - 9.85e7T + 5.55e15T^{2} \) |
| 5 | \( 1 - 3.74e11T + 1.16e23T^{2} \) |
| 7 | \( 1 + 1.82e13T + 7.73e27T^{2} \) |
| 11 | \( 1 + 2.87e17T + 2.32e34T^{2} \) |
| 13 | \( 1 - 3.77e18T + 5.75e36T^{2} \) |
| 17 | \( 1 - 2.29e20T + 4.02e40T^{2} \) |
| 19 | \( 1 + 1.83e20T + 1.58e42T^{2} \) |
| 23 | \( 1 + 4.16e22T + 8.65e44T^{2} \) |
| 29 | \( 1 - 8.55e23T + 1.81e48T^{2} \) |
| 31 | \( 1 + 2.36e24T + 1.64e49T^{2} \) |
| 37 | \( 1 + 8.47e24T + 5.63e51T^{2} \) |
| 41 | \( 1 + 5.60e26T + 1.66e53T^{2} \) |
| 43 | \( 1 + 9.34e25T + 8.02e53T^{2} \) |
| 47 | \( 1 - 3.13e27T + 1.51e55T^{2} \) |
| 53 | \( 1 + 3.41e27T + 7.96e56T^{2} \) |
| 59 | \( 1 + 5.18e28T + 2.74e58T^{2} \) |
| 61 | \( 1 - 2.71e29T + 8.23e58T^{2} \) |
| 67 | \( 1 + 2.81e29T + 1.82e60T^{2} \) |
| 71 | \( 1 - 4.32e30T + 1.23e61T^{2} \) |
| 73 | \( 1 + 1.93e30T + 3.08e61T^{2} \) |
| 79 | \( 1 + 8.03e30T + 4.18e62T^{2} \) |
| 83 | \( 1 + 5.17e31T + 2.13e63T^{2} \) |
| 89 | \( 1 - 1.98e32T + 2.13e64T^{2} \) |
| 97 | \( 1 - 2.61e32T + 3.65e65T^{2} \) |
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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
−20.53058432789469914546486500619, −18.46052819043271712489008514299, −15.80122209652185855259865700661, −14.02982978564107361760562836626, −13.16845213122273547496469714219, −10.16076196227441268183542950897, −8.145887549812245160496356076384, −5.72735934513992236494353208315, −3.30274134422974868219869663836, −1.99487032221332408813705573070,
1.99487032221332408813705573070, 3.30274134422974868219869663836, 5.72735934513992236494353208315, 8.145887549812245160496356076384, 10.16076196227441268183542950897, 13.16845213122273547496469714219, 14.02982978564107361760562836626, 15.80122209652185855259865700661, 18.46052819043271712489008514299, 20.53058432789469914546486500619
